Complete randomized design without treatment structure

observational units were randomly assigned to the treatments - only systematic influence on the outcome should be due to the treatment - treatment effect

1. first impression of the data

data("InsectSprays")
with(InsectSprays, tapply(count, spray, length))
##  A  B  C  D  E  F 
## 12 12 12 12 12 12
with(InsectSprays, tapply(count, spray, summary))
## $A
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    7.00   11.50   14.00   14.50   17.75   23.00 
## 
## $B
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    7.00   12.50   16.50   15.33   17.50   21.00 
## 
## $C
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   1.000   1.500   2.083   3.000   7.000 
## 
## $D
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.000   3.750   5.000   4.917   5.000  12.000 
## 
## $E
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    1.00    2.75    3.00    3.50    5.00    6.00 
## 
## $F
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    9.00   12.50   15.00   16.67   22.50   26.00
with(InsectSprays, tapply(count, spray, sd))
##        A        B        C        D        E        F 
## 4.719399 4.271115 1.975225 2.503028 1.732051 6.213378

2. Visualization

stripplots for less than 15 points per treatment interpretation: higher or lower values for the dependent variable? variance?

library(ggplot2)
ggplot(InsectSprays, aes(spray, count))+geom_point(shape=1, position=position_jitter(width=0.1, height=0))

library(lattice)
stripplot(count~spray, data=InsectSprays, jitter=0.2)

boxplots Interpretation: median differences? variance? - compare box heights skewness? - whiskers and median outliers? - data points outside the whiskers

plot(count~spray, data=InsectSprays)

Another visualization: Both plots contain the same information the means +- their standard error is plotted

library(sciplot)
bargraph.CI(spray, count, col=(gray(0.88)), data=InsectSprays, xlab='spray', ylab='count', ylim=c(0,20))

lineplot.CI(spray, count, type='p', data=InsectSprays, xlab='spray', ylab='count', ylim=c(0,20))

InsectSprays$block=factor(rep(rep(1:6, each=2), times=6))
ggplot(InsectSprays, aes(spray, count)) + geom_point() + facet_grid(~block)

s.o <- with(InsectSprays, reorder(spray, count, mean, order = TRUE))
InsectSprays$spray.o <- factor(InsectSprays$spray, levels = levels(s.o))
b.o <- with(InsectSprays, reorder(block, count, mean, order = TRUE))
InsectSprays$block.o <- factor(InsectSprays$block, levels = levels(b.o))
ggplot(InsectSprays, aes(spray.o, count)) + geom_point() + facet_grid( ~ block.o) + xlab("spray")

visualization accounting for block structure the second plot orders the blocks and treatments by count means asses: do blocks themselves have an effect? do treatment effects depend on the blocks? here no block effects visible

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Parametric models: one-way ANOVA

Assumptions: independence, equal variance, normal distributed measurements for each treatment follow a normal distribution with mean \(\mu\) and variance \(\sigma\)2, assumed to be fixed unkown numbers with the variance being the same for each treatment.

ins.lm = lm(count~spray, data=InsectSprays)
anova(ins.lm)
#same results:
ins.aov = aov(count~spray, data=InsectSprays)
summary(ins.aov)
##             Df Sum Sq Mean Sq F value Pr(>F)    
## spray        5   2669   533.8    34.7 <2e-16 ***
## Residuals   66   1015    15.4                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

If the p-value of the overall F test is smaller than the significance level, reject the null hypothesis and conclude that not all the treatments have the same theoretical means. Only valid if model assumptions hold.

spray - Sum Sq: SSB, if SSB=0 (i.e. SST=SSW) all sample treatments means are equal to the overall mean. If the null hypothesis is true SSB should be small otherwise it should be big. spray - Mean Sq: MSB=SSB/(k-1) quantifies the variability between the treatment means. Residuals - Sum Sq: SSW, if SSW=0 (i.e. SST=SSB) within every treatment, all the values are equal to the respective mean.

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Residuals, RMSE, and the R2

residuals := observed values - fitted values RMSE := root mean squared error (estimates the standard deviation \(\sigma\) of our observations) R2 := 1-SSW/SST (the coefficient of determination: is the proportion of variance of the dependent variable explained by the model)

# RMSE
summary(ins.lm)$sigma
## [1] 3.921902
#R^2^
summary(ins.lm)$r.squared
## [1] 0.724439

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Estimating and testing group means - cell means model

ins.lm.noint = lm(count~spray-1, data=InsectSprays)
summary(ins.lm.noint)$coefficients
##         Estimate Std. Error   t value     Pr(>|t|)
## sprayA 14.500000   1.132156 12.807428 1.470512e-19
## sprayB 15.333333   1.132156 13.543487 1.001994e-20
## sprayC  2.083333   1.132156  1.840148 7.024334e-02
## sprayD  4.916667   1.132156  4.342749 4.953047e-05
## sprayE  3.500000   1.132156  3.091448 2.916794e-03
## sprayF 16.666667   1.132156 14.721181 1.573471e-22

excludes intercept since we do not have a control treatment, shows two-sided t test wether the respective group mean is zero.

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Pairwise treatment comparison

the multiple testing problem If we perform a family of hypothesis tests (example compare pairwise k treatments: m = k*(k-1)/2 hypothesis tests). The probability of committing at least one Type I error in all the tests together is called the family error rate (1-(1-\(\alpha\))m). The increase of the familywise error rate is called the multiple testing problem.

#with(InsectSprays, pairwise.t.test(count, spray, 'holm'))
#with(InsectSprays, pairwise.wilcox.test(count, spray, 'holm'))

Gives adjusted p-values so that the family error rate is 0.05. Commented here since it gives too many warnings..

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Tukey’s honest significance difference (HSD)

Pairwise comparison of treatment means. assumptions: normality, roughly similar variances sensitive to outliers and conservative if not all treatments have the same number of observations.

TukeyHSD(ins.aov, 'spray')
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = count ~ spray, data = InsectSprays)
## 
## $spray
##            diff        lwr       upr     p adj
## B-A   0.8333333  -3.866075  5.532742 0.9951810
## C-A -12.4166667 -17.116075 -7.717258 0.0000000
## D-A  -9.5833333 -14.282742 -4.883925 0.0000014
## E-A -11.0000000 -15.699409 -6.300591 0.0000000
## F-A   2.1666667  -2.532742  6.866075 0.7542147
## C-B -13.2500000 -17.949409 -8.550591 0.0000000
## D-B -10.4166667 -15.116075 -5.717258 0.0000002
## E-B -11.8333333 -16.532742 -7.133925 0.0000000
## F-B   1.3333333  -3.366075  6.032742 0.9603075
## D-C   2.8333333  -1.866075  7.532742 0.4920707
## E-C   1.4166667  -3.282742  6.116075 0.9488669
## F-C  14.5833333   9.883925 19.282742 0.0000000
## E-D  -1.4166667  -6.116075  3.282742 0.9488669
## F-D  11.7500000   7.050591 16.449409 0.0000000
## F-E  13.1666667   8.467258 17.866075 0.0000000
library(agricolae)
HSD.test(ins.lm, 'spray', group=TRUE, console=TRUE)
## 
## Study: ins.lm ~ "spray"
## 
## HSD Test for count 
## 
## Mean Square Error:  15.38131 
## 
## spray,  means
## 
##       count      std  r Min Max
## A 14.500000 4.719399 12   7  23
## B 15.333333 4.271115 12   7  21
## C  2.083333 1.975225 12   0   7
## D  4.916667 2.503028 12   2  12
## E  3.500000 1.732051 12   1   6
## F 16.666667 6.213378 12   9  26
## 
## Alpha: 0.05 ; DF Error: 66 
## Critical Value of Studentized Range: 4.150851 
## 
## Minimun Significant Difference: 4.699409 
## 
## Treatments with the same letter are not significantly different.
## 
##       count groups
## F 16.666667      a
## B 15.333333      a
## A 14.500000      a
## D  4.916667      b
## E  3.500000      b
## C  2.083333      b

either values with p adj < \(\alpha\) or different letters are significantly different

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Completely randomized designs with control

Comparing all treatments with the control

We want to test if the treatment mean is different from the control treatment mean

data("InsectSprays")
ins.lm = lm(count~spray, data=InsectSprays)
summary(ins.lm)$coefficient
##                Estimate Std. Error    t value     Pr(>|t|)
## (Intercept)  14.5000000   1.132156 12.8074279 1.470512e-19
## sprayB        0.8333333   1.601110  0.5204724 6.044761e-01
## sprayC      -12.4166667   1.601110 -7.7550382 7.266893e-11
## sprayD       -9.5833333   1.601110 -5.9854322 9.816910e-08
## sprayE      -11.0000000   1.601110 -6.8702352 2.753922e-09
## sprayF        2.1666667   1.601110  1.3532281 1.805998e-01
contrasts(InsectSprays$spray)
##   B C D E F
## A 0 0 0 0 0
## B 1 0 0 0 0
## C 0 1 0 0 0
## D 0 0 1 0 0
## E 0 0 0 1 0
## F 0 0 0 0 1
with(InsectSprays, tapply(count, spray, mean))
##         A         B         C         D         E         F 
## 14.500000 15.333333  2.083333  4.916667  3.500000 16.666667

the intercept estimates the mean of the reference level while all other coefficients estimate differences of the respective treatment to the mean of the reference treatment. Estimated mean insects surviving a treatment with reference level (spray A) is 14.5. For the reference level the t test measures if the mean is zero, for the other treatments it measures if the treatment mean is different from the reference mean.

Contrasts & Sum Contrasts

options('contrasts')
## $contrasts
##         unordered           ordered 
## "contr.treatment"      "contr.poly"

Sometimes we want to know if the treatment differs from the grand mean, the average of all treatment means.

ins.lm.sum = lm(count~spray, data=InsectSprays, contrast=list(spray='contr.sum'))
summary(ins.lm.sum)$coefficients
##              Estimate Std. Error   t value     Pr(>|t|)
## (Intercept)  9.500000  0.4622006 20.553848 2.160853e-30
## spray1       5.000000  1.0335119  4.837874 8.223823e-06
## spray2       5.833333  1.0335119  5.644186 3.778197e-07
## spray3      -7.416667  1.0335119 -7.176180 7.866853e-10
## spray4      -4.583333  1.0335119 -4.434718 3.571783e-05
## spray5      -6.000000  1.0335119 -5.805449 2.003785e-07
InsectSprays$spray.sum = InsectSprays$spray
contrasts(InsectSprays$spray.sum) = 'contr.sum'
contrasts(InsectSprays$spray.sum)
##   [,1] [,2] [,3] [,4] [,5]
## A    1    0    0    0    0
## B    0    1    0    0    0
## C    0    0    1    0    0
## D    0    0    0    1    0
## E    0    0    0    0    1
## F   -1   -1   -1   -1   -1

The intercept is now an estimate for the overall mean. The contrast matrix shows that spray F is now interpreted as the reference level.

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Specific comparisons

Reducing the number of comparisons is attractive since it makes the p value correction less strict, such that we gain in power.

The multcomp library includes:

  • Dunnet - Compare all levels to a reference level
  • GrandMean - Compare all levels to the overall mean
  • Sequen - Test a specific sequence
  • Tukey - Tukey’s HSD
library(multcomp)
## Loading required package: mvtnorm
## Loading required package: survival
## Loading required package: TH.data
## Loading required package: MASS
## 
## Attaching package: 'TH.data'
## The following object is masked from 'package:MASS':
## 
##     geyser
summary(glht(ins.lm, mcp(spray='Dunnett')))
## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Multiple Comparisons of Means: Dunnett Contrasts
## 
## 
## Fit: lm(formula = count ~ spray, data = InsectSprays)
## 
## Linear Hypotheses:
##            Estimate Std. Error t value Pr(>|t|)    
## B - A == 0   0.8333     1.6011   0.520    0.979    
## C - A == 0 -12.4167     1.6011  -7.755   <1e-04 ***
## D - A == 0  -9.5833     1.6011  -5.985   <1e-04 ***
## E - A == 0 -11.0000     1.6011  -6.870   <1e-04 ***
## F - A == 0   2.1667     1.6011   1.353    0.526    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)

p values are adjusted. Reference level is A.

Test specific hypotheses: pass contrast matrix Here we compare the means of treatment spray B and treatment spray F.

K = matrix(c(0,1,0,0,0,-1), nrow=1)
summary(glht(ins.lm, linfct=K))
## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Fit: lm(formula = count ~ spray, data = InsectSprays)
## 
## Linear Hypotheses:
##        Estimate Std. Error t value Pr(>|t|)
## 1 == 0   -1.333      1.601  -0.833    0.408
## (Adjusted p values reported -- single-step method)
summary(glht(ins.lm, mcp(spray = 'GrandMean')))
## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Multiple Comparisons of Means: GrandMean Contrasts
## 
## 
## Fit: lm(formula = count ~ spray, data = InsectSprays)
## 
## Linear Hypotheses:
##        Estimate Std. Error t value Pr(>|t|)    
## A == 0    5.000      1.034   4.838  < 1e-04 ***
## B == 0    5.833      1.034   5.644  < 1e-04 ***
## C == 0   -7.417      1.034  -7.176  < 1e-04 ***
## D == 0   -4.583      1.034  -4.435 0.000218 ***
## E == 0   -6.000      1.034  -5.805  < 1e-04 ***
## F == 0    7.167      1.034   6.934  < 1e-04 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)

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Model diagnostics and alternative analysis methods

Normality

Graphical normaility checking

Check if the residuals come from a normal distribution by comparing the quantiles of the residual distribution with the quantiles of a standard normal distribution. Do not use for less than 30 samples.

library(car)
## Loading required package: carData
qqPlot(resid(ins.lm))

## [1] 69 70

Testing for normality - Shapiro Wilk

Reject the null hypothesis of normality if p < \(\alpha\)

shapiro.test(resid(ins.lm))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(ins.lm)
## W = 0.96006, p-value = 0.02226

For small samples a nonsignificant Shapiro Wilk test does not imply no problems with normality are present.

Accounting for nonnormality

  • transformation g of Y, so that the transfromation g(Y) produces better behaved residuals. Very useful especially when the transformation has a nice interpretation. - Box Cox transformation
  • abandon normal distributed based methods use non parametric, robust, bootstrap or exact methods instead.

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Homoskedasicity

Graphical homoskedasticity checking - residual plots

plot(fitted(ins.lm), resid(ins.lm), las=1, xlab='Fitted Values', ylab='Residuals')
abline(h=0)

plot(ins.lm, which=1)

The variance of the residuals seems to increase with the fitted values, we conclude that the assumption of equal variance is violkated.

Testing for homoskedasticity

high variance implies a low reliability

bartlett.test(count~spray,  data=InsectSprays)
## 
##  Bartlett test of homogeneity of variances
## 
## data:  count by spray
## Bartlett's K-squared = 25.96, df = 5, p-value = 9.085e-05
leveneTest(count~spray, data=InsectSprays)

if p < \(\alpha\) reject the null hypothesis of equal variance.

Accounting for heteroskedasticity

  • find a transformation g such that the variances are more homogenous
  • use robust methods
  • perform a one way ANOVA with adjusted degrees of freedom, can be used if there are no problems with normality
oneway.test(count~spray, data=InsectSprays)
## 
##  One-way analysis of means (not assuming equal variances)
## 
## data:  count and spray
## F = 36.065, num df = 5.000, denom df = 30.043, p-value = 7.999e-12

Violations of the model assumption can very severely inflate the probability of a Type I error, i.e. that the F test rejects the true null hypothesis of no treatment effects too often on average) and diminish the power, i.e. that the F test too often fails to reject false null hypothesis.

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Nonparametric models

Assumptions:

  1. the n random variables are mutually independent
  2. all observations from treatment j come from the same continuous distribution F_j
  3. the treatment distribution function F_j are shifted varaints of some distribution F

The parametric model is retrieved if one assumes that F is a normal distribution function

The Kruskal-Wallis test

Tests the null hypothesis that each of the underlying distributions is the same against the alternative that at least two treatment effects differ. If the null hypothesis is true mean ranks should not differ too much between treatments - has problems with ties! Suffers if the variances are not equal.

kruskal.test(count~spray, data=InsectSprays)
## 
##  Kruskal-Wallis rank sum test
## 
## data:  count by spray
## Kruskal-Wallis chi-squared = 54.691, df = 5, p-value = 1.511e-10

To test which treatments differ from each other, we apply a special ad hoc procedures to perform all two-sided pairwise comparisons.

library(NSM3)
## Loading required package: combinat
## 
## Attaching package: 'combinat'
## The following object is masked from 'package:utils':
## 
##     combn
## Loading required package: partitions
## 
## Attaching package: 'partitions'
## The following object is masked from 'package:car':
## 
##     S
#with(InsectSprays, pSDCFlig(x=count, g=as.numeric(spray), method=NA))

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Robust methods

The Brunner Dette Munk test

Not affected as much if we do not have equal variance. Tests the nullhypotheis that the distributions of the numeric variable are the same in each treatment.

library(asbio)
## Loading required package: tcltk
## Registered S3 method overwritten by 'asbio':
##   method   from
##   print.ci coin
with(InsectSprays, BDM.test(count,spray))
## 
## One way Brunner-Dette-Munk test 
## 
##       df1      df2       F*    P(F > F*)
##  4.828351 63.19189 44.26642 4.138278e-19

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The trim test

useful for data with outliers and tests the null hypothesis of equal trimmed means instead of medians.

library(asbio)
with(InsectSprays, trim.test(count, spray, tr=0.2))
## $Results
##   df1      df2       F*        P(>F)
## 1   5 18.92297 28.22811 3.745716e-08

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Permutation methods

library(lmPerm)
anova(lmp(count~spray, data=InsectSprays))
## [1] "Settings:  unique SS "
## Analysis of Variance Table
## 
## Response: count
##           Df R Sum Sq R Mean Sq Iter  Pr(Prob)    
## spray      5   2668.8    533.77 5000 < 2.2e-16 ***
## Residuals 66   1015.2     15.38                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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Blocking

Blocking: grouping of homogeneous units to remove block-to-block variation from the experimental error. 1. Natural discrete blocks: The plots may naturally be grouped in blocks. example: piglets from the same mother

  1. Continuous gradients The plots may show differences which seem to change along a continous gradient. This is often due to plot homogeneity in space or time. example: agricultural fields

  2. Trial management blocks Many experiments need more staff than one person, trials might be executed differently. If possible staff assignment should reflect staff assignment.

Principles for blocking: 1. all blocks should have the same size 2. be sufficiently big to apply each treatment at least once per block

Complete block designs

In the randomized complete block design (RCBD), every treatment is applied in every plot exactly one time, allocating treatments randomly within blocks.

In the generalized complete block design (GRCBD) every treatment us applied in every block exactly r > 1 times, allocating treatments randomly within blocks.

Fixed block effects model

Only takes the fixed effects of the blocks and the treatments into account, treatment effects are the same in each block. There is no block x treatment interaction. Usually not what we want: 1. The interest is usually not in the fixed effect of the block 2. estimation of the block effects takes too many degrees of freedom 3. we do not care if the block effect is significant, we only want to correctly estimate the treatment effect while accounting for the plot structure induced by the blocks. 4. The assumption that the error terms are independent might not be particularly realistic.

Random block effect model - mixed effect model

Accounts for the random block effects. The main interest is not in the effect of the blocks, only in the variation introduced by the blocking factor. We have a fixed treatment effect and two random components the block effect and the error terms, both normally distributed with mean zero.

Do not ignore block effects! Otherwise the variability due to block effects falls into the errors standard deviation making it much harder to detect the real treatment effect!

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Fitting and interpreting mixed effects models

The data set must have one variable for the blocks - long format

InsectSprays$block = factor(rep(rep(1:6, each=2), times=6))
head(InsectSprays)
library(nlme)
ins.lme = lme(sqrt(count)~spray, random=~1|block, data=InsectSprays)

One random effect, a random intercept per block. For the InsectSprays data we use a square root transformation for the data. The REML (restricted maximum likelihood) is the default of lme, and is used to get better variance estimates.

summary(ins.lme)
## Linear mixed-effects model fit by REML
##  Data: InsectSprays 
##        AIC      BIC    logLik
##   152.2289 169.7461 -68.11445
## 
## Random effects:
##  Formula: ~1 | block
##         (Intercept) Residual
## StdDev:   0.2540835 0.579766
## 
## Fixed effects: sqrt(count) ~ spray 
##                 Value Std.Error DF    t-value p-value
## (Intercept)  3.760678 0.1969021 61  19.099225  0.0000
## sprayB       0.115953 0.2366885 61   0.489897  0.6260
## sprayC      -2.515822 0.2366885 61 -10.629254  0.0000
## sprayD      -1.596325 0.2366885 61  -6.744412  0.0000
## sprayE      -1.951217 0.2366885 61  -8.243821  0.0000
## sprayF       0.257939 0.2366885 61   1.089782  0.2801
##  Correlation: 
##        (Intr) sprayB sprayC sprayD sprayE
## sprayB -0.601                            
## sprayC -0.601  0.500                     
## sprayD -0.601  0.500  0.500              
## sprayE -0.601  0.500  0.500  0.500       
## sprayF -0.601  0.500  0.500  0.500  0.500
## 
## Standardized Within-Group Residuals:
##        Min         Q1        Med         Q3        Max 
## -2.4815817 -0.5213930 -0.1471345  0.6234485  1.9749960 
## 
## Number of Observations: 72
## Number of Groups: 6
fixef(ins.lme)
## (Intercept)      sprayB      sprayC      sprayD      sprayE      sprayF 
##   3.7606784   0.1159530  -2.5158217  -1.5963245  -1.9512174   0.2579388

Gives: 1. standard deviation of the blocks and the residuals, shows if the block effect is negligible compared to the errors. 2. AIC and BIC criterion, smaller values are preferable 3. REML estimates of the foxed effects with the usual t tests and the correlations of the fixed effects estimator.

The random effects are not estimated but predicted, one for each block.

ranef(ins.lme)
ranef(ins.lme)$'(Intercept)' 
## [1] -0.30503308  0.28296568 -0.01645621 -0.10988229  0.19387989 -0.04547399

The random effects show how a particular block compares to a ‘typical’ block with random effect of zero. Negative values mean that the block has less insects than a typical block, positive values mean higher counts.

Compare block sample means to the overall mean

blk.centred = with(InsectSprays, tapply(sqrt(count), block, mean)-mean(sqrt(count)))
blk.ranef = ranef(ins.lme)$'(Intercept)'
cbind(blk.centred, blk.ranef, ratio = blk.centred/blk.ranef)
##   blk.centred   blk.ranef    ratio
## 1 -0.43738127 -0.30503308 1.433881
## 2  0.40573924  0.28296568 1.433881
## 3 -0.02359626 -0.01645621 1.433881
## 4 -0.15755818 -0.10988229 1.433881
## 5  0.27800078  0.19387989 1.433881
## 6 -0.06520431 -0.04547399 1.433881

All the random effects predictions are essentially the deviations of the block means from the overall mean, but shrunk towards zero by the same factor.

within group fitted values include random effects or level to zero to only include fixed effects

head(fitted(ins.lme))
##        1        1        2        2        3        3 
## 3.455645 3.455645 4.043644 4.043644 3.744222 3.744222
head(fitted(ins.lme, level=0))
##        1        1        2        2        3        3 
## 3.760678 3.760678 3.760678 3.760678 3.760678 3.760678

The same applies to residuals.

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Model diagnostics

Equal variance

plot(ins.lme)

Does the variance of the error terms change with the fitted values? Wedge shape? The lme function can handle modelled different variances, for example different variances according to the levels of the factor.

Normality

qqPlot(resid(ins.lme))

##  5  6 
## 34 23
shapiro.test(resid(ins.lme))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(ins.lme)
## W = 0.98985, p-value = 0.8351
shapiro.test(ranef(ins.lme)$'(Intercept)')
## 
##  Shapiro-Wilk normality test
## 
## data:  ranef(ins.lme)$"(Intercept)"
## W = 0.96421, p-value = 0.8515

Typically testing for normality of the random effect of the blocks does not make sense since we have less than 30 blocks and the tests are not meaningful.

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Confidence intervals and hypothesis test

Give a first impression about the precision of the estimators, but are sensitiv to outliers and ill-behaved data sets (nonnormality, unequal variances)

intervals(ins.lme)
## Approximate 95% confidence intervals
## 
##  Fixed effects:
##                  lower       est.      upper
## (Intercept)  3.3669482  3.7606784  4.1544086
## sprayB      -0.3573348  0.1159530  0.5892408
## sprayC      -2.9891095 -2.5158217 -2.0425339
## sprayD      -2.0696124 -1.5963245 -1.1230367
## sprayE      -2.4245052 -1.9512174 -1.4779296
## sprayF      -0.2153491  0.2579388  0.7312266
## attr(,"label")
## [1] "Fixed effects:"
## 
##  Random Effects:
##   Level: block 
##                     lower      est.     upper
## sd((Intercept)) 0.1041355 0.2540835 0.6199464
## 
##  Within-group standard error:
##     lower      est.     upper 
## 0.4855018 0.5797660 0.6923322
anova(ins.lme, type='marginal')

Test if all fxed effects estimates except the intercept are zero. If the null hypothesis is rejected we get a significant effect of the treatments (we do not know which one).

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Treatment comparisons

We can set contrasts and use the multcomp library to make pairwise comparisons.

library(multcomp)
summary(glht(ins.lme, mcp(spray='GrandMean')))
## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Multiple Comparisons of Means: GrandMean Contrasts
## 
## 
## Fit: lme.formula(fixed = sqrt(count) ~ spray, data = InsectSprays, 
##     random = ~1 | block)
## 
## Linear Hypotheses:
##        Estimate Std. Error z value Pr(>|z|)    
## A == 0   0.9482     0.1528   6.207  < 1e-04 ***
## B == 0   1.0642     0.1528   6.965  < 1e-04 ***
## C == 0  -1.5676     0.1528 -10.260  < 1e-04 ***
## D == 0  -0.6481     0.1528  -4.242  0.00017 ***
## E == 0  -1.0030     0.1528  -6.565  < 1e-04 ***
## F == 0   1.2062     0.1528   7.895  < 1e-04 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
summary(glht(ins.lme, mcp(spray='Tukey')))
## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Multiple Comparisons of Means: Tukey Contrasts
## 
## 
## Fit: lme.formula(fixed = sqrt(count) ~ spray, data = InsectSprays, 
##     random = ~1 | block)
## 
## Linear Hypotheses:
##            Estimate Std. Error z value Pr(>|z|)    
## B - A == 0   0.1160     0.2367   0.490  0.99654    
## C - A == 0  -2.5158     0.2367 -10.629  < 0.001 ***
## D - A == 0  -1.5963     0.2367  -6.744  < 0.001 ***
## E - A == 0  -1.9512     0.2367  -8.244  < 0.001 ***
## F - A == 0   0.2579     0.2367   1.090  0.88568    
## C - B == 0  -2.6318     0.2367 -11.119  < 0.001 ***
## D - B == 0  -1.7123     0.2367  -7.234  < 0.001 ***
## E - B == 0  -2.0672     0.2367  -8.734  < 0.001 ***
## F - B == 0   0.1420     0.2367   0.600  0.99107    
## D - C == 0   0.9195     0.2367   3.885  0.00152 ** 
## E - C == 0   0.5646     0.2367   2.385  0.16114    
## F - C == 0   2.7738     0.2367  11.719  < 0.001 ***
## E - D == 0  -0.3549     0.2367  -1.499  0.66468    
## F - D == 0   1.8543     0.2367   7.834  < 0.001 ***
## F - E == 0   2.2092     0.2367   9.334  < 0.001 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)

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Using lsmeans/emmeans

Used for comparing treatments especially in more complex settings estimated marginal means: The idea is to define a reference grid of interesting values (for factors these are just their levels) and then compute the fitted values according to the mdoel on the reference grid.

library(emmeans)
ref_grid(ins.lme)
## 'emmGrid' object with variables:
##     spray = A, B, C, D, E, F
## Transformation: "sqrt"
ins.emm = emmeans(ins.lme,'spray')
ins.emm
##  spray emmean    SE df lower.CL upper.CL
##  A       3.76 0.197  5    3.255     4.27
##  B       3.88 0.197  5    3.370     4.38
##  C       1.24 0.197  5    0.739     1.75
##  D       2.16 0.197  5    1.658     2.67
##  E       1.81 0.197  5    1.303     2.32
##  F       4.02 0.197  5    3.512     4.52
## 
## Degrees-of-freedom method: containment 
## Results are given on the sqrt (not the response) scale. 
## Confidence level used: 0.95
pairs(ins.emm)
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
##  contrast estimate    SE df t.ratio p.value
##  A - B      -0.116 0.237 61  -0.490 0.9964 
##  A - C       2.516 0.237 61  10.629 <.0001 
##  A - D       1.596 0.237 61   6.744 <.0001 
##  A - E       1.951 0.237 61   8.244 <.0001 
##  A - F      -0.258 0.237 61  -1.090 0.8836 
##  B - C       2.632 0.237 61  11.119 <.0001 
##  B - D       1.712 0.237 61   7.234 <.0001 
##  B - E       2.067 0.237 61   8.734 <.0001 
##  B - F      -0.142 0.237 61  -0.600 0.9907 
##  C - D      -0.919 0.237 61  -3.885 0.0033 
##  C - E      -0.565 0.237 61  -2.385 0.1776 
##  C - F      -2.774 0.237 61 -11.719 <.0001 
##  D - E       0.355 0.237 61   1.499 0.6659 
##  D - F      -1.854 0.237 61  -7.834 <.0001 
##  E - F      -2.209 0.237 61  -9.334 <.0001 
## 
## Note: contrasts are still on the sqrt scale 
## Degrees-of-freedom method: containment 
## P value adjustment: tukey method for comparing a family of 6 estimates
plot(ins.emm, comparison=TRUE)
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates

The blue bars of the plots depicts confidence intervals of the marginal means. The red arrows are for the pairwise comparisons among the spray means. If any arrow from one spray overlaps with an arrow from another spray, their mean difference is not significant.

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Ignoring blocks

If we do not account for the block structure the estimate for of the errors standard deviation is bigger in the model without block effects, since the variability of the blocks is not properly accounted for.

The intraclass correlation coefficient

Observations from different blocks are independent according to the definition of a mixed model. Observations from the same block are correlated because they share the block random effect. All observations in the same block share the same correlation given by the intraclass correlation coefficient. (block variance over sum block variance and error variance)

V = VarCorr(ins.lme)
V = as.numeric(V)
V[1]/(V[1]+V[2])
## [1] 0.1611194

Due to the definition used here negative correlations are not possible and can only be attained if the model is defined in a different way. Negative correlation can be expected if there is competition inside blocks for example for food and light.

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The coefficient of determiniation

Because we have to account for the random effect of the blocks the decomposition of the sum of squares no longer works as it does in linear models. As a result no values for R2 are generally given. If we want to quantify the fit of the model we can give the marginal and conditional R2 values extracted as follows.

library(MuMIn)
r.squaredGLMM(ins.lme)
## Warning: 'r.squaredGLMM' now calculates a revised statistic. See the help page.
##            R2m       R2c
## [1,] 0.7566123 0.7958267

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An example: the immer data and the nonparametric Friedman test

The immer data contains the yield of five varieties (M, P, S, T and V) of barley grown in six locations Loc (C, D, GR, M, UF and W) for two years (Y1 and Y2).

library(MASS)
data(immer)
immer
ggplot(immer, aes(x=Var, y=Y1)) + geom_point() + facet_grid(~Loc)

The data inside the locations is quite homogeneous but the locations greatly differ from each other. If we ignoring the locations will result in pooling heterogeneous data together. This will be reflected in the residual variance which will be overestimated and we do not get a significant treatment (variety) effect. If we inlcude the location (blocks) as a fixed effect we do not account for the randomization which happens inside locations. Since we are not interested in the fixed effect but rather in the random effect introduced trough the location we can treat it as a blocking factor. (Only include locations as a fixed effect if you are excactly interested in those locations.)

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A parametric analysis - include the location as a random effect

immer.lme = lme(Y1~Var, random=~1|Loc, data=immer)
immer.lme
## Linear mixed-effects model fit by REML
##   Data: immer 
##   Log-restricted-likelihood: -111.3314
##   Fixed: Y1 ~ Var 
## (Intercept)        VarP        VarS        VarT        VarV 
## 102.5833333   7.1666667  -0.5500000  24.8166667   0.8833333 
## 
## Random effects:
##  Formula: ~1 | Loc
##         (Intercept) Residual
## StdDev:    26.08863 12.76273
## 
## Number of Observations: 30
## Number of Groups: 6
anova(immer.lme)

We have a significant location effect. The standard deviation of the random intercepts is even bigger than the residual standard deviation, because the blocks are so different. The only drawback of this analysis is that the sample is so small and it is very difficult asses the model assumptions in a reliable way. We could use a non parametric analysis instead.

plot(immer.lme)

qqPlot(resid(immer.lme))

## UF  M 
##  3 14
shapiro.test(resid(immer.lme))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(immer.lme)
## W = 0.96611, p-value = 0.4389

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A non parametric analysis - Friedman’s test

The Friedman’s test is a non parametric test of the null hypothesis that the treatment effect is the same for every level of the treatment in RCBD. The test idea is to rank observations within blocks and then sum ranks for the treatments over all the blocks. The ranking makes the method robust to outliers - this robustness is paid for some loss of power (compared to the parametric model) if the data do come from a normal distribution. Assumptions: the distribution of the dependent variables for the levels of the treatment are shifted variants of the same distribution. The test should not be used for widely different distributions or if the treatments work differently for some block, i.e. if we have treatment x block interactions.

friedman.test(Y1~Var|Loc, immer)
## 
##  Friedman rank sum test
## 
## data:  Y1 and Var and Loc
## Friedman chi-squared = 10.933, df = 4, p-value = 0.02732

We get a significant variety effect.

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Interactions of block effects and treatment effects

If we have GRCBD (more than one observation per treatment per block) we can answer the question whether the treatment effects are different by block. Visualization by plotting means for each combination of block and treatments interaction plot

with(InsectSprays, interaction.plot(spray, block, sqrt(count)))

No interaction: the traces should be more or less parallel. Approach: fit a model with nested random effects. The blocks are a random sample from the set of all possible blocks, so that any differences in treatment effects due to the blocks should also be treated as random effects. We can fit a model with two levels of random effects: one level for the blocks and one level for the treatments within each block.

ins.lme.x = lme(sqrt(count)~spray, random=~1|block/spray, data=InsectSprays)
ins.lme.x
## Linear mixed-effects model fit by REML
##   Data: InsectSprays 
##   Log-restricted-likelihood: -67.43423
##   Fixed: sqrt(count) ~ spray 
## (Intercept)      sprayB      sprayC      sprayD      sprayE      sprayF 
##   3.7606784   0.1159530  -2.5158217  -1.5963245  -1.9512174   0.2579388 
## 
## Random effects:
##  Formula: ~1 | block
##         (Intercept)
## StdDev:   0.2394903
## 
##  Formula: ~1 | spray %in% block
##         (Intercept)  Residual
## StdDev:   0.2706021 0.5254596
## 
## Number of Observations: 72
## Number of Groups: 
##            block spray %in% block 
##                6               36

The log-likelihood of the more complex model is slightly higher, is the use of the more complex model justified? Since we fitted the model with the maximum likelihood method we use the likelihood-ratio test to quantify whether the improvement of the likelihood is sufficient given the additional model complexity.

anova(ins.lme, ins.lme.x)

Give the simpler model first. The simpler model seems to be sufficient, i.e. we do not have any evidence for an interaction of treatment effects and blocks. Thus we should proceed with the simpler model.

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Design with factorial treatment structure

The turnip yield data The question is how the planting density in kg/ha affects the mean yield of turnips of two genotypes. Turn planting density in a factor first.

library(agridat)
turnip = mcconway.turnip
turnip$density = as.factor(turnip$density)

ggplot(turnip, aes(x = density, y = yield)) + geom_point() + facet_grid(~gen)

head(turnip)
tail(turnip)
with(turnip, tapply(yield, list(gen, density), mean))
##              1   2      4      8
## Barkant 2.9375 4.4 9.0875 9.6625
## Marco   1.3375 2.3 5.5625 7.7250

Visualization and means of every combination of density and genotype.

Questions: 1. Is there a planting density effect? - look at the visualization 2. Is there a genotype effect vary among the two genotype?

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Visualization

qplot(density, yield, data=turnip, facets=~block, shape=gen, size=I(2))

Block differences are clearly visible, Block 4 seems to have really low yield, and we can see an overall increase of the yield with the planting density.

Interaction plot

ggplot(turnip, aes(x=density, linetype=gen, group=gen, y=yield)) + stat_summary(fun.y=mean, geom='point') + stat_summary(fun.y=mean, geom='line')
## Warning: `fun.y` is deprecated. Use `fun` instead.

## Warning: `fun.y` is deprecated. Use `fun` instead.

Do the lines look more or less parallel? Here they do.. no interaction effect

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A model for factorial two-way ANOVA with replications - balanced design

We want to study the influence of two factors on a numeric variable. We are interested in the fixed effects of both factors. Assumption: data come from a normal distribution with an equal variance for each group.

The effect model

In comparison to the cell means model which does only account for finding effects of the two factors combined, we can use the factor effect model two find separate main effects of the two factors and the interaction effect of them.

turnip.full = lm(yield~gen*density, data=turnip)
coef(summary(turnip.full))
##                   Estimate Std. Error    t value    Pr(>|t|)
## (Intercept)         2.9375   1.522377  1.9295486 0.058734960
## genMarco           -1.6000   2.152966 -0.7431609 0.460490758
## density2            1.4625   2.152966  0.6792955 0.499748857
## density4            6.1500   2.152966  2.8565245 0.005999627
## density8            6.7250   2.152966  3.1235980 0.002827723
## genMarco:density2  -0.5000   3.044754 -0.1642169 0.870151687
## genMarco:density4  -1.9250   3.044754 -0.6322351 0.529806221
## genMarco:density8  -0.3375   3.044754 -0.1108464 0.912134467

R automatically chooses a reference level. The intercept is the estimated yield for the Barkant (reference level) and density 1. The yield at density 1 for the Marco variety is estimated as (2.9375-1.6000). If the interaction effect is bigger than zero, than the combined effect of one level of factor and one level of factor Z is bigger than the sum of the main effects; this is called a positive interaction. If the interaction effect is smaller than zero, this is called a negative interaction. For a balanced design, the combination of the above estimates yield exactly the sample means.

with(turnip, tapply(yield, list(gen, density), mean))
##              1   2      4      8
## Barkant 2.9375 4.4 9.0875 9.6625
## Marco   1.3375 2.3 5.5625 7.7250

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F tests

The overall F test

Is the model using two factors and their interaction effect any better than predicting the overall mean for every observation (null model)?

summary(turnip.full)
## 
## Call:
## lm(formula = yield ~ gen * density, data = turnip)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.2875 -2.5500 -0.2187  1.8156 14.9125 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)   
## (Intercept)         2.9375     1.5224   1.930  0.05873 . 
## genMarco           -1.6000     2.1530  -0.743  0.46049   
## density2            1.4625     2.1530   0.679  0.49975   
## density4            6.1500     2.1530   2.857  0.00600 **
## density8            6.7250     2.1530   3.124  0.00283 **
## genMarco:density2  -0.5000     3.0448  -0.164  0.87015   
## genMarco:density4  -1.9250     3.0448  -0.632  0.52981   
## genMarco:density8  -0.3375     3.0448  -0.111  0.91213   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.306 on 56 degrees of freedom
## Multiple R-squared:  0.3516, Adjusted R-squared:  0.2705 
## F-statistic: 4.338 on 7 and 56 DF,  p-value: 0.0006739

The overall F test is in the last line and is significant, one should not proceed with partial F tests if the overall F test is not significant.

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The ANOVA table and the F tests for main and interaction effects

Do we need the interaction of the two factors? Test the null hypothesis that all interaction effects are zero. Do we need a main factor? Test the null hypothesis that the coefficients of the main effect are zero.

anova(turnip.full)

First check if the interaction effect is significant only remove non-significant interaction effects. If the interaction effect is significant, do not remove the involved main effects. If the interaction effect is non-significant one can remove main effects if they are also non-significant to simplify the model.

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A model with only the main effects and not the interaction effects

turnip.main = lm(yield~gen + density, data=turnip)

Tests on individual coefficients

summary(turnip.full)
## 
## Call:
## lm(formula = yield ~ gen * density, data = turnip)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.2875 -2.5500 -0.2187  1.8156 14.9125 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)   
## (Intercept)         2.9375     1.5224   1.930  0.05873 . 
## genMarco           -1.6000     2.1530  -0.743  0.46049   
## density2            1.4625     2.1530   0.679  0.49975   
## density4            6.1500     2.1530   2.857  0.00600 **
## density8            6.7250     2.1530   3.124  0.00283 **
## genMarco:density2  -0.5000     3.0448  -0.164  0.87015   
## genMarco:density4  -1.9250     3.0448  -0.632  0.52981   
## genMarco:density8  -0.3375     3.0448  -0.111  0.91213   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.306 on 56 degrees of freedom
## Multiple R-squared:  0.3516, Adjusted R-squared:  0.2705 
## F-statistic: 4.338 on 7 and 56 DF,  p-value: 0.0006739

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Fitted values and confidence intervals

dat = expand.grid(gen=levels(turnip$gen), density=levels(turnip$density))
pred = predict(object=turnip.sqrt, newdata=dat, interval='confidence')
ci = cbind(dat, pred^2)
ci

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Post hoc tests

Suppose we want to compare the yield of the two genotypes. You could now average over the planting densities, which should only be done if the genotype effect is more or less the same for each planting density (check interaction plot and interaction effect).

emmeans(turnip.sqrt, pairwise~gen)
## NOTE: Results may be misleading due to involvement in interactions
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
## $emmeans
##  gen     emmean    SE df lower.CL upper.CL
##  Barkant   2.34 0.144 56     2.05     2.63
##  Marco     1.86 0.144 56     1.57     2.15
## 
## Results are averaged over the levels of: density 
## Results are given on the sqrt (not the response) scale. 
## Confidence level used: 0.95 
## 
## $contrasts
##  contrast        estimate    SE df t.ratio p.value
##  Barkant - Marco    0.479 0.203 56 2.358   0.0219 
## 
## Results are averaged over the levels of: density 
## Note: contrasts are still on the sqrt scale

On average, over all the planting densities, the Barnkant genotype produces significantly higher yields. If we do have interactions it would be better to compare the genotypes separately for each planting density.

turnip.sqrt.gen = emmeans(turnip.sqrt, pairwise~gen|density)
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
turnip.sqrt.gen$contrasts
## density = 1:
##  contrast        estimate    SE df t.ratio p.value
##  Barkant - Marco    0.490 0.407 56 1.204   0.2336 
## 
## density = 2:
##  contrast        estimate    SE df t.ratio p.value
##  Barkant - Marco    0.557 0.407 56 1.371   0.1759 
## 
## density = 4:
##  contrast        estimate    SE df t.ratio p.value
##  Barkant - Marco    0.539 0.407 56 1.325   0.1906 
## 
## density = 8:
##  contrast        estimate    SE df t.ratio p.value
##  Barkant - Marco    0.332 0.407 56 0.817   0.4176 
## 
## Note: contrasts are still on the sqrt scale

Because the standard errors are now higher, the separate comparison no longer produce significant differences. This is why you should exclude non-significant interactions from a model. The model should not be more complex than it should have to be. We could also compare all the combinations of genotype and planting density pairwisely:

emmeans(turnip.sqrt, pairwise~gen+density)
## Note: Use 'contrast(regrid(object), ...)' to obtain contrasts of back-transformed estimates
## $emmeans
##  gen     density emmean    SE df lower.CL upper.CL
##  Barkant 1         1.61 0.288 56    1.038     2.19
##  Marco   1         1.12 0.288 56    0.548     1.70
##  Barkant 2         2.02 0.288 56    1.448     2.60
##  Marco   2         1.47 0.288 56    0.890     2.04
##  Barkant 4         2.74 0.288 56    2.168     3.32
##  Marco   4         2.20 0.288 56    1.629     2.78
##  Barkant 8         2.98 0.288 56    2.403     3.55
##  Marco   8         2.65 0.288 56    2.071     3.22
## 
## Results are given on the sqrt (not the response) scale. 
## Confidence level used: 0.95 
## 
## $contrasts
##  contrast              estimate    SE df t.ratio p.value
##  Barkant 1 - Marco 1     0.4896 0.407 56  1.204  0.9273 
##  Barkant 1 - Barkant 2  -0.4099 0.407 56 -1.008  0.9713 
##  Barkant 1 - Marco 2     0.1475 0.407 56  0.363  1.0000 
##  Barkant 1 - Barkant 4  -1.1297 0.407 56 -2.778  0.1214 
##  Barkant 1 - Marco 4    -0.5910 0.407 56 -1.453  0.8280 
##  Barkant 1 - Barkant 8  -1.3649 0.407 56 -3.357  0.0288 
##  Barkant 1 - Marco 8    -1.0329 0.407 56 -2.540  0.2005 
##  Marco 1 - Barkant 2    -0.8995 0.407 56 -2.212  0.3605 
##  Marco 1 - Marco 2      -0.3421 0.407 56 -0.841  0.9898 
##  Marco 1 - Barkant 4    -1.6194 0.407 56 -3.982  0.0046 
##  Marco 1 - Marco 4      -1.0806 0.407 56 -2.658  0.1577 
##  Marco 1 - Barkant 8    -1.8545 0.407 56 -4.561  0.0007 
##  Marco 1 - Marco 8      -1.5225 0.407 56 -3.744  0.0095 
##  Barkant 2 - Marco 2     0.5574 0.407 56  1.371  0.8664 
##  Barkant 2 - Barkant 4  -0.7198 0.407 56 -1.770  0.6421 
##  Barkant 2 - Marco 4    -0.1811 0.407 56 -0.445  0.9998 
##  Barkant 2 - Barkant 8  -0.9550 0.407 56 -2.349  0.2867 
##  Barkant 2 - Marco 8    -0.6230 0.407 56 -1.532  0.7869 
##  Marco 2 - Barkant 4    -1.2772 0.407 56 -3.141  0.0509 
##  Marco 2 - Marco 4      -0.7385 0.407 56 -1.816  0.6120 
##  Marco 2 - Barkant 8    -1.5124 0.407 56 -3.719  0.0103 
##  Marco 2 - Marco 8      -1.1804 0.407 56 -2.903  0.0913 
##  Barkant 4 - Marco 4     0.5387 0.407 56  1.325  0.8855 
##  Barkant 4 - Barkant 8  -0.2352 0.407 56 -0.578  0.9990 
##  Barkant 4 - Marco 8     0.0969 0.407 56  0.238  1.0000 
##  Marco 4 - Barkant 8    -0.7739 0.407 56 -1.903  0.5546 
##  Marco 4 - Marco 8      -0.4419 0.407 56 -1.087  0.9570 
##  Barkant 8 - Marco 8     0.3320 0.407 56  0.817  0.9915 
## 
## Note: contrasts are still on the sqrt scale 
## P value adjustment: tukey method for comparing a family of 8 estimates

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Unbalanced design

Problem: the sums of squares no longer add up as they do in the balanced case. Consequence: Distinguish between sequential and marginal F tests.

Marginal F tests asses whether an effect significantly reduces the resdiual sum of squares adjusted for all the other effects at the same or lower level. The order of terms in the model equation does not matter.

library(car)
Anova(turnip.sqrt, type=2)

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Accounting for blocks and for heteroscedasticity

How to account for the blocks of the turnip data? Use a linear mixed-effect model with random intercept per block:

turnip.lme = lme(sqrt(yield)~gen*density, random=~1|block, data=turnip)

This model combines the factorial treatment structure with the random intercept per block.

Diagnostics

plot(turnip.lme)

qqPlot(resid(turnip.lme))

## B1 B4 
## 25 12
shapiro.test(resid(turnip.lme))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(turnip.lme)
## W = 0.98437, p-value = 0.595

Normaility looks good, but the variance still increases with the fitted values.

Heteroscedastic within-group errors Estimate a separate variance for each level of a factor. We specify that the mixed model should estimate a separate variance for each level of density.

turnip.lme.het = lme(sqrt(yield)~gen*density, random=~1|block, data=turnip, weights=varIdent(form=~1|density))
turnip.lme.het
## Linear mixed-effects model fit by REML
##   Data: turnip 
##   Log-restricted-likelihood: -67.33769
##   Fixed: sqrt(yield) ~ gen * density 
##       (Intercept)          genMarco          density2          density4 
##        1.61393154       -0.48964237        0.40986169        1.12970894 
##          density8 genMarco:density2 genMarco:density4 genMarco:density8 
##        1.36488918       -0.06774089       -0.04907923        0.15760443 
## 
## Random effects:
##  Formula: ~1 | block
##         (Intercept)  Residual
## StdDev:   0.2332858 0.4805598
## 
## Variance function:
##  Structure: Different standard deviations per stratum
##  Formula: ~1 | density 
##  Parameter estimates:
##         1         2         4         8 
## 1.0000000 0.9609899 2.1941207 1.8086482 
## Number of Observations: 64
## Number of Groups: 4
plot(turnip.lme.het)

The unequal variance problem seems to be completely solved. We can now use this model to proceed with the analysis.

anova(turnip.lme.het, type='marginal')

Conclude that the interaction effect is not needed and simplify the model:

turnip.lme.het.add = lme(sqrt(yield)~gen+density, random=~1|block, data=turnip, weights=varIdent(form=~1|density))

anova(turnip.lme.het.add)

Both main effects are sigificant we could proceed with further analysis!

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Classical analysis (ANOVA tables)

Complete blocked designs

If we have designs with plot structure, we have to tell the aov() function about the plot structure otherwise the wrong F tests are produced.

summary(aov(sqrt(count)~spray+Error(block), data=InsectSprays))
## 
## Error: block
##           Df Sum Sq Mean Sq F value Pr(>F)
## Residuals  5  5.554   1.111               
## 
## Error: Within
##           Df Sum Sq Mean Sq F value Pr(>F)    
## spray      5  88.44  17.688   52.62 <2e-16 ***
## Residuals 61  20.50   0.336                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The Error term is used to tell the aov() to add a random block effect, as a result no test for the significance of the block effect is performed, as this is usually not of interest. Or we could fit with lme and get the same results.

library(nlme)
ins.lme=lme(sqrt(count)~spray, random=~1|block, data=InsectSprays)
anova(ins.lme, type='marginal')

Use aov() just for balanced designs and lme() for a more general approach.

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Balanced incomplete block design

If block sizes are smaller than the number of treatments, complete block designs are not realizable. Instead we use a binary design, this means in each block one treatment is either observed once or not at all. And all treatments are observed the same number of times.

Definition: Each of the v treatments is observed r times , and each of the b blocks has k plots (uniformity). Also the design is binary and each pair of treatments appears together \(\lambda\) times (number of concurrences).

Conditions (necessary not sufficient): 1. The amount of treatments times the amount of times each treatment is observed is equal to the amount of blocks times the amount of plots: v\(\times\)r = b\(\times\)k 2. r(k-1) = \(\lambda\)\(\times\)(v-1), each treatment is applied in r blocks and in each of these blocks are k-1 other treatments. Since each of the treatments is exactly observed \(\lambda\) times together with of the v-1 other treatments, the condition follows. 3. b \(\le\) v

library(ibd)
set.seed(1)
bibd(7, 7, 3, 3, 1)$design 
##         [,1] [,2] [,3]
## Block-1    1    4    6
## Block-2    5    6    7
## Block-3    3    4    5
## Block-4    1    3    7
## Block-5    2    4    7
## Block-6    2    3    6
## Block-7    1    2    5

We have bibd(v, b, r, k, lambda, ntrail, pbar=FALSE)

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Example: The detergent data

detergent = read.table('detergent.csv', header=TRUE, sep=',')
head(detergent)
addmargins(with(detergent, table(treatment, block))) #shows design
##          block
## treatment B01 B02 B03 B04 B05 B06 B07 B08 B09 B10 B11 B12 Sum
##       T1    0   0   0   1   0   0   0   1   1   0   0   1   4
##       T2    0   1   0   0   1   0   0   0   0   1   0   1   4
##       T3    1   0   1   0   0   0   0   0   0   0   1   1   4
##       T4    1   1   0   0   0   1   0   1   0   0   0   0   4
##       T5    0   0   0   1   0   1   0   0   0   1   1   0   4
##       T6    0   0   1   0   1   1   0   0   1   0   0   0   4
##       T7    0   0   0   0   1   0   1   1   0   0   1   0   4
##       T8    1   0   0   0   0   0   1   0   1   1   0   0   4
##       T9    0   1   1   1   0   0   1   0   0   0   0   0   4
##       Sum   3   3   3   3   3   3   3   3   3   3   3   3  36
ggplot(detergent, aes(treatment, plates)) + geom_jitter(shape=4, width=0.1)

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Fixed effects model - intrablock analysis

We have a fixed treatment effect, a fixed block effect and a random error term. The model assumes no interaction effects between block and treatments. Question: Is there a treatment effect after adjusting for any block effect? (The order of the variables is important, block comes first for this analysis!)

detergent.lm = lm(plates~block+treatment, data=detergent)
anova(detergent.lm)

This strategy is sometimes called intrablock analysis because we base the analysis on differences from within the block. The treatment effect adjusted for blocks is highly significant, now we could proceed with pairwise comparisons or treatment contrasts.

Adjusting for block effects Adjust the observation by the difference of the least square estimate of the effect of the block and the least squares estimate of the mean block effect.

theta.hat = c(0, coef(detergent.lm)[2:12])
theta.avg = mean(theta.hat)
detergent$block.num = as.factor(detergent$block)
detergent$adj = theta.hat[detergent$block.num] - theta.avg
detergent$plates.adj = detergent$plates - detergent$adj
head(detergent)
ggplot(detergent, aes(treatment, plates.adj)) + geom_jitter(shape=4, width=0.1)

The adjusted values do not differ a lot from the original values, since the treatment effects dominate block effect for the data set.

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Mixed effects model - interblock effect analysis

Block effects are modeled as random effects.

detergent.lme = lme(plates~treatment, random=~1|block, data=detergent)
anova(detergent.lme)

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Split-plot designs - the Oats example

Visualization of the raw data

#split for each block, colors show variety
ggplot(Oats, aes(x=nitro, y=yield, col=Variety)) + geom_point() + geom_line() + facet_wrap(~Block)

#split for each variety, grouped according to block
ggplot(Oats, aes(x=nitro, y=yield, group=Block)) + geom_point() + geom_line() + facet_wrap(~Variety)

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Basic statistical model

Classically, split plot designs are analyzed with nested random effects. The default treatment reference levels are the first levels of both factors. We have the two fixed main effects of the two factors and their interaction effect. A random intercept for each block, as well as a random intercept for each plot within the block and a random error term.

We assume that the random effects and the error terms are independent.

library(nlme)
Oats$nitroF = factor(Oats$nitro) #turn the nitrogen levels into factors
Oats.lme = lme(yield~Variety*nitroF, random=~1|Block/Variety, data=Oats) 
summary(Oats.lme)
## Linear mixed-effects model fit by REML
##  Data: Oats 
##        AIC      BIC    logLik
##   559.0285 590.4437 -264.5143
## 
## Random effects:
##  Formula: ~1 | Block
##         (Intercept)
## StdDev:    14.64496
## 
##  Formula: ~1 | Variety %in% Block
##         (Intercept) Residual
## StdDev:    10.29863 13.30727
## 
## Fixed effects: yield ~ Variety * nitroF 
##                                Value Std.Error DF   t-value p-value
## (Intercept)                 80.00000  9.106958 45  8.784492  0.0000
## VarietyMarvellous            6.66667  9.715028 10  0.686222  0.5082
## VarietyVictory              -8.50000  9.715028 10 -0.874933  0.4021
## nitroF0.2                   18.50000  7.682957 45  2.407927  0.0202
## nitroF0.4                   34.66667  7.682957 45  4.512152  0.0000
## nitroF0.6                   44.83333  7.682957 45  5.835427  0.0000
## VarietyMarvellous:nitroF0.2  3.33333 10.865342 45  0.306786  0.7604
## VarietyVictory:nitroF0.2    -0.33333 10.865342 45 -0.030679  0.9757
## VarietyMarvellous:nitroF0.4 -4.16667 10.865342 45 -0.383482  0.7032
## VarietyVictory:nitroF0.4     4.66667 10.865342 45  0.429500  0.6696
## VarietyMarvellous:nitroF0.6 -4.66667 10.865342 45 -0.429500  0.6696
## VarietyVictory:nitroF0.6     2.16667 10.865342 45  0.199411  0.8428
##  Correlation: 
##                             (Intr) VrtyMr VrtyVc ntF0.2 ntF0.4 ntF0.6 VM:F0.2
## VarietyMarvellous           -0.533                                           
## VarietyVictory              -0.533  0.500                                    
## nitroF0.2                   -0.422  0.395  0.395                             
## nitroF0.4                   -0.422  0.395  0.395  0.500                      
## nitroF0.6                   -0.422  0.395  0.395  0.500  0.500               
## VarietyMarvellous:nitroF0.2  0.298 -0.559 -0.280 -0.707 -0.354 -0.354        
## VarietyVictory:nitroF0.2     0.298 -0.280 -0.559 -0.707 -0.354 -0.354  0.500 
## VarietyMarvellous:nitroF0.4  0.298 -0.559 -0.280 -0.354 -0.707 -0.354  0.500 
## VarietyVictory:nitroF0.4     0.298 -0.280 -0.559 -0.354 -0.707 -0.354  0.250 
## VarietyMarvellous:nitroF0.6  0.298 -0.559 -0.280 -0.354 -0.354 -0.707  0.500 
## VarietyVictory:nitroF0.6     0.298 -0.280 -0.559 -0.354 -0.354 -0.707  0.250 
##                             VV:F0.2 VM:F0.4 VV:F0.4 VM:F0.6
## VarietyMarvellous                                          
## VarietyVictory                                             
## nitroF0.2                                                  
## nitroF0.4                                                  
## nitroF0.6                                                  
## VarietyMarvellous:nitroF0.2                                
## VarietyVictory:nitroF0.2                                   
## VarietyMarvellous:nitroF0.4  0.250                         
## VarietyVictory:nitroF0.4     0.500   0.500                 
## VarietyMarvellous:nitroF0.6  0.250   0.500   0.250         
## VarietyVictory:nitroF0.6     0.500   0.250   0.500   0.500 
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -1.81300898 -0.56144838  0.01758044  0.63864476  1.57034166 
## 
## Number of Observations: 72
## Number of Groups: 
##              Block Variety %in% Block 
##                  6                 18

The standard deviation of the block is given with 14.645, for the variety inside the blocks with 10.298 and for the error terms with 13.307. To test the significance of the fixed effects we are using the marginal F test approach.

anova(Oats.lme, type='marginal')

The interaction effect is not significant! No variety specific effect of the nitrogen level can be detected. We can remove the interaction effect and fit the model again.

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Model reduction

Fit the model again without the non significant interaction effect.

Oats.lme = lme(yield~Variety+nitroF, random=~1|Block/Variety, data=Oats)
summary(Oats.lme)
## Linear mixed-effects model fit by REML
##  Data: Oats 
##        AIC      BIC    logLik
##   586.0688 605.7756 -284.0344
## 
## Random effects:
##  Formula: ~1 | Block
##         (Intercept)
## StdDev:    14.64488
## 
##  Formula: ~1 | Variety %in% Block
##         (Intercept) Residual
## StdDev:    10.47345 12.74986
## 
## Fixed effects: yield ~ Variety + nitroF 
##                      Value Std.Error DF   t-value p-value
## (Intercept)       79.91667  8.220351 51  9.721807  0.0000
## VarietyMarvellous  5.29167  7.078910 10  0.747526  0.4720
## VarietyVictory    -6.87500  7.078910 10 -0.971195  0.3544
## nitroF0.2         19.50000  4.249955 51  4.588284  0.0000
## nitroF0.4         34.83333  4.249955 51  8.196166  0.0000
## nitroF0.6         44.00000  4.249955 51 10.353051  0.0000
##  Correlation: 
##                   (Intr) VrtyMr VrtyVc ntF0.2 ntF0.4
## VarietyMarvellous -0.431                            
## VarietyVictory    -0.431  0.500                     
## nitroF0.2         -0.259  0.000  0.000              
## nitroF0.4         -0.259  0.000  0.000  0.500       
## nitroF0.6         -0.259  0.000  0.000  0.500  0.500
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -1.84134146 -0.66279747 -0.06694274  0.63822469  1.66066801 
## 
## Number of Observations: 72
## Number of Groups: 
##              Block Variety %in% Block 
##                  6                 18
anova(Oats.lme, type='marginal')

The F test shows that the main effect of the Variety is not significant. This does not mean that we completely remove the Variety from the model, we remove the main effect but still keep it as a random effect, since it models intra-strip correlation within each block. The fixed effect models a systematic variety difference across blocks since it is not specific to a particular block.

Fit the model again without the main effect for Variety:

Oats.lme = lme(yield~nitroF, random=~1|Block/Variety, data=Oats)
summary(Oats.lme)
## Linear mixed-effects model fit by REML
##  Data: Oats 
##        AIC      BIC    logLik
##   596.2187 611.7553 -291.1094
## 
## Random effects:
##  Formula: ~1 | Block
##         (Intercept)
## StdDev:    14.50594
## 
##  Formula: ~1 | Variety %in% Block
##         (Intercept) Residual
## StdDev:    11.03866 12.74987
## 
## Fixed effects: yield ~ nitroF 
##                Value Std.Error DF   t-value p-value
## (Intercept) 79.38889  7.132392 51 11.130753       0
## nitroF0.2   19.50000  4.249957 51  4.588282       0
## nitroF0.4   34.83333  4.249957 51  8.196161       0
## nitroF0.6   44.00000  4.249957 51 10.353046       0
##  Correlation: 
##           (Intr) ntF0.2 ntF0.4
## nitroF0.2 -0.298              
## nitroF0.4 -0.298  0.500       
## nitroF0.6 -0.298  0.500  0.500
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -1.78155507 -0.61168963  0.02222435  0.62200776  1.68137700 
## 
## Number of Observations: 72
## Number of Groups: 
##              Block Variety %in% Block 
##                  6                 18
anova(Oats.lme)

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Exploiting order structure

So far we ignored the ordered structure of the nitrogen factor. Since all levels are equidistant, we can treat the levels as numeric. To take advantage of the ordered structure we could convert the nitrogen into an ordered factor. Here we fit the model again with no fixed effect for Variety:

Oats$nitroF = factor(Oats$nitro, ordered=TRUE)

Oats.lme.o = lme(yield~nitroF, random=~1|Block/Variety, data=Oats)
coef(summary(Oats.lme.o))
##                   Value Std.Error DF    t-value      p-value
## (Intercept) 103.9722222  6.640669 51 15.6568891 4.030058e-21
## nitroF.L     32.9447349  3.005169 51 10.9626912 5.121203e-15
## nitroF.Q     -5.1666667  3.005169 51 -1.7192602 9.163171e-02
## nitroF.C     -0.4472136  3.005169 51 -0.1488148 8.822866e-01

If we use an ordered structure so called orthogonal polynomial contrasts are used for the nitrogen factor. The first contrast estimates the linear trend, the second the quadratic effect orthogonal (independent) of the linear trend, and the third estimates the orthogonal cubic trend. We can see that only the linear trend is significant. We want to look again at the fitted values (excluding random effects) from the model that treats nitrogen as a factor without ordered structure:

df = data.frame(nitroF=levels(Oats$nitroF))
df$pred = predict(Oats.lme, df, level=0)
df

Here the linearity seem to be reflected except maybe for the last level. We can now fit our last model with a linear regression structure for the fixed effect of nitrogen:

Oats.lme.lin = lme(yield~nitro, random=~1|Block/Variety, data=Oats)
summary(Oats.lme.lin)
## Linear mixed-effects model fit by REML
##  Data: Oats 
##        AIC      BIC    logLik
##   603.0418 614.2842 -296.5209
## 
## Random effects:
##  Formula: ~1 | Block
##         (Intercept)
## StdDev:    14.50598
## 
##  Formula: ~1 | Variety %in% Block
##         (Intercept) Residual
## StdDev:    11.00467 12.86696
## 
## Fixed effects: yield ~ nitro 
##                Value Std.Error DF  t-value p-value
## (Intercept) 81.87222  6.945280 53 11.78818       0
## nitro       73.66667  6.781483 53 10.86291       0
##  Correlation: 
##       (Intr)
## nitro -0.293
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -1.74380777 -0.66475224  0.01710434  0.54298790  1.80298914 
## 
## Number of Observations: 72
## Number of Groups: 
##              Block Variety %in% Block 
##                  6                 18

We can now interpret that an increase of the nitrogen level of 0.2 units rises the yield by 0.2$$73.66 = 14.73 units on average for a typical plot. Visualization:

Oats$yield.factor = predict(Oats.lme)
Oats$yield.lin = predict(Oats.lme.lin)
ggplot(Oats, aes(x = nitro, y = yield, col = Variety)) + geom_point() + geom_line(linetype = "solid") + geom_line(aes(y = yield.factor, col = Variety), linetype = "dotted") + geom_line(aes(y = yield.lin, col = Variety), linetype = "dashed") + facet_wrap(~Block)

The factor fitted model is represented by the solid lines, the linear regression model fitted values by dashed lines. The two models seem to make very similar predictions for the fitted values, if that is the case the linear model is to be preferred since less parameters have to be estimated (bias-variance trade off).

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Using covariate information - the Gasel example

We have five blocks depending on the distance to a hedge, in each block one plot is close to the forest and one further away (the variable location acts as a treatment here). One person should do the measurements for one block, to cancel out the observer bias. Strictly speaking this is not an experiment, because of the missing randomization, but an observational study. We want to examine the location effect (treatment) on the shannon index.

gasel = read.table('gasel.csv', sep=',', header=TRUE)
head(gasel)
ggplot(gasel, aes(block, shannon, col=location)) + geom_point()

ggplot(gasel, aes(location, shannon, col=location)) + geom_point() + facet_grid(~block)

The Shannon index (which measures biodiversity) is higher in open plots. Is the effect significant? Since we do not have many samples a non parametric test seems to be the right choice for analysation of the data.

friedman.test(shannon~location|block, data=gasel)
## 
##  Friedman rank sum test
## 
## data:  shannon and location and block
## Friedman chi-squared = 5, df = 1, p-value = 0.02535

The Friedmans test finds a significant treatment, here location, effect. To use covariates we use a parametric mixed effects model (although the data sample is perhaps too small).

gasel.blk = lme(shannon~location, random=~1|block, data=gasel)
coef(summary(gasel.blk))
##                 Value  Std.Error DF   t-value      p-value
## (Intercept)     2.768 0.11968713  4 23.126965 0.0000207149
## locationshaded -0.444 0.07978719  4 -5.564803 0.0051073584

The mixed effects model also shows a significant location effect (open plots are the reference level) and we get an estimate for the effect. For a typical block the Shannon index is of the open plots is 2.77 and the Shannon is on average 0.44 units lower in the shaded plots than in the open plots.

We now want to also take into account the effect of the radiation (Wh/m2) data on the Shannon index. Visualization:

gasel$radiation = gasel$radiation/100000
ggplot(gasel, aes(radiation, shannon)) + geom_point(aes(col=location)) + geom_line(aes(group=block))

The two plots of each block are connected by line. In each block a clear effect of radiation is seen. We now fit a model and allow two different lienar regression lines to be fit for shaded and open plots:

gasel.lme = lme(shannon~radiation*location, random=~1|block, data=gasel)
coef(summary(gasel.lme))
##                                Value Std.Error DF    t-value   p-value
## (Intercept)               1.95111780 1.5909478  4  1.2263871 0.2873091
## radiation                 0.20860938 0.4053724  2  0.5146117 0.6580502
## locationshaded           -0.26602504 1.4582940  2 -0.1824221 0.8720680
## radiation:locationshaded  0.04623994 0.3653919  2  0.1265489 0.9108725
ggplot(gasel, aes(radiation, shannon, col=location)) + geom_point() + geom_smooth(method='lm', se=FALSE)
## `geom_smooth()` using formula 'y ~ x'

The slope for the shaded plots is 0.046 units higher than the slope of the open locations, but the slope difference is not significant (interaction effect). So we can fit a model without it, but still allow for different intercepts:

gasel.lme = lme(shannon~radiation+location, random=~1 | block, data=gasel)
coef(summary(gasel.lme))
##                     Value Std.Error DF    t-value    p-value
## (Intercept)     1.8180334 0.6181093  4  2.9412813 0.04233508
## radiation       0.2425955 0.1557267  3  1.5578284 0.21715256
## locationshaded -0.1022203 0.2366051  3 -0.4320293 0.69488937
ggplot(gasel, aes(radiation, shannon, col=location)) + geom_point() + geom_smooth(aes(y = predict(gasel.lme, gasel)), method='lm', se=FALSE)
## `geom_smooth()` using formula 'y ~ x'

The shaded locations have an intercept which is 0.1 units lower than the one for the open locations, but again the difference is not significant. Further reduction to just one common regression line:

gasel.lme = lme(shannon~radiation, random=~1|block, data=gasel)
coef(summary(gasel.lme))
##                 Value  Std.Error DF  t-value     p-value
## (Intercept) 1.5625107 0.20123014  4 7.764794 0.001482769
## radiation   0.3062472 0.05719127  4 5.354789 0.005866770
fixef(gasel.lme)
## (Intercept)   radiation 
##   1.5625107   0.3062472
ggplot(gasel, aes(radiation, shannon)) + geom_point() + geom_smooth(aes(y=predict(gasel.lme, gasel)), method='lm', se=FALSE)
## `geom_smooth()` using formula 'y ~ x'

For this data set the covariate was so important that it completely eliminated the treatment from the model.

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Local Teaching Session 2 - the PlantGrowth data (one-way ANOVA)

data("PlantGrowth")
head(PlantGrowth)

1. Use suitable descriptive statistics to describe the data set

with(PlantGrowth, tapply(weight, group, length))
## ctrl trt1 trt2 
##   10   10   10
with(PlantGrowth, tapply(weight, group, summary))
## $ctrl
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   4.170   4.550   5.155   5.032   5.293   6.110 
## 
## $trt1
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   3.590   4.207   4.550   4.661   4.870   6.030 
## 
## $trt2
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   4.920   5.268   5.435   5.526   5.735   6.310
with(PlantGrowth, tapply(weight, group, sd))
##      ctrl      trt1      trt2 
## 0.5830914 0.7936757 0.4425733

Each of the three treatments has the same amount of observations, 10 per treatment. The range of the values is the biggest for treatment 1, which has the smallest mean with 4.66 and the highest standard deviation with 0.79, the control treatment has a mean of 5.03 and a standard deviation of 0.58, the treatment 2 has a mean of 5.52 and a standard deviation of 0.44.

2. Visualize the data suitably. What do you observe?

library(ggplot2)
ggplot(PlantGrowth, aes(group, weight)) + geom_jitter(width=0.1, height=0)

We obtain a strip plot of the data. We can see that the group of trt1 has two data points which are quite far away from the other data points, this also explains the high standard deviation.

3. Perform the parametric F test and interpret the result.

plant.lm = lm(weight~group, data=PlantGrowth)
anova(plant.lm)

The parametric overall F test indicates a significant treatment effect, i.e. the null hypothesis of equal group means can be rejected for a significance level of \(\alpha\) = 0.05, with a p-value of 0.0159.

4. Compare the three groups using Tukey’s HSD (not multcomp)

plant.aov = aov(weight~group, data=PlantGrowth)
TukeyHSD(plant.aov)
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = weight ~ group, data = PlantGrowth)
## 
## $group
##             diff        lwr       upr     p adj
## trt1-ctrl -0.371 -1.0622161 0.3202161 0.3908711
## trt2-ctrl  0.494 -0.1972161 1.1852161 0.1979960
## trt2-trt1  0.865  0.1737839 1.5562161 0.0120064

We do not have a significant difference for the control treatment mean to either of the two treatment means, but we do have a significant difference between the two treatment means (p-value = 0.012). With this method of pairwise comparisons the familywise error rate is conrtolled at 5%.

5. Now use the multcomp package to perform Tukey’s HSD. Compare with the previous problem.

library(multcomp)
summary(glht(plant.lm, mcp(group='Tukey')))
## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Multiple Comparisons of Means: Tukey Contrasts
## 
## 
## Fit: lm(formula = weight ~ group, data = PlantGrowth)
## 
## Linear Hypotheses:
##                  Estimate Std. Error t value Pr(>|t|)  
## trt1 - ctrl == 0  -0.3710     0.2788  -1.331    0.391  
## trt2 - ctrl == 0   0.4940     0.2788   1.772    0.198  
## trt2 - trt1 == 0   0.8650     0.2788   3.103    0.012 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)

The HSD test of the multcomp package also just finds a significant difference in means between treatment 1 and treatment 2. We can also say that trt2 produces a weight which is on average 0.865 units higher than trt1.

6. Use the multcomp package to compare each of the two treatments to the control only (but not each other). Compare the results with the previous problem.

summary(glht(plant.lm, mcp(group='Dunnet')))
## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Multiple Comparisons of Means: Dunnett Contrasts
## 
## 
## Fit: lm(formula = weight ~ group, data = PlantGrowth)
## 
## Linear Hypotheses:
##                  Estimate Std. Error t value Pr(>|t|)
## trt1 - ctrl == 0  -0.3710     0.2788  -1.331    0.323
## trt2 - ctrl == 0   0.4940     0.2788   1.772    0.153
## (Adjusted p values reported -- single-step method)

As before we do not have a significant difference for the treatment means compared to the control treatment. We can see that the p-values are a bit smaller than for Tukey’s HSD since the multiplicity adjustment is more strict for three test than for two tests.

7. Judge resiudal normality visually and with statistical test. What is your conclusion?

library(car)
qqPlot(resid(plant.lm))

## [1] 17 15
shapiro.test(resid(plant.lm))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(plant.lm)
## W = 0.96607, p-value = 0.4379

Visually we do have a slightly heavy upper tail, but this could also just be due to chance. The null hypothesis of normality is not rejected by the shapiro test, so we do not detected any problems with non normality.

8. Do the residuals have equal variances?

plot(fitted(plant.lm), resid(plant.lm), las=1, xlab='Fitted values', ylab='Residuals')
abline(h=0)

#or 

plot(plant.lm, which=1)

The variance of the residuals seems to decrease with an increase of the fitted values. To check the variance further we could perform a test.

bartlett.test(weight~group, data=PlantGrowth)
## 
##  Bartlett test of homogeneity of variances
## 
## data:  weight by group
## Bartlett's K-squared = 2.8786, df = 2, p-value = 0.2371

The Bartlett does not reject the null hypothesis of equal variance (p-value = 0.2371), so we conclude that there a no problems with heteroscedasticity.

9. Perform and interpret a Kruskal-Wallis test for the PlantGrowth data.

kruskal.test(weight~group, data=PlantGrowth)
## 
##  Kruskal-Wallis rank sum test
## 
## data:  weight by group
## Kruskal-Wallis chi-squared = 7.9882, df = 2, p-value = 0.01842

The Kruskal Wallis test is significant, that means that the null hypothesis of equal underlying weight distributions is rejected and at least two treatment effects differ. In conclusion a significant effect (p-value = 0.01842) of the treatment on teh weight distribution is found.

10. Use a robust method on this data set and interpret the result.

library(asbio)
with(PlantGrowth, BDM.test(weight, group))
## 
## One way Brunner-Dette-Munk test 
## 
##       df1     df2       F*  P(F > F*)
##  1.873924 23.7978 5.132413 0.01546714
#or

with(PlantGrowth, trim.test(weight, group))
## $Results
##   df1      df2       F*       P(>F)
## 1   2 9.639328 8.372075 0.007806059

The Brunner-Dette-Munkrejects the null hypothesis that the distribution of the weights are the same in each group (p-value = 0.0155).

The Trim test finds a significant difference of the 20% trimmed means in the three groups (p-value = 0.0078).

11. Use a permutation test to test for equality of the three treatment means.

library(lmPerm)
set.seed(42)
anova(lmp(weight~group, data=PlantGrowth))
## [1] "Settings:  unique SS "
## Analysis of Variance Table
## 
## Response: weight
##           Df R Sum Sq R Mean Sq Iter Pr(Prob)   
## group      2   3.7663    1.8832 5000    0.009 **
## Residuals 27  10.4921    0.3886                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The Permutation test has a significant Monte Carlo p-value with 0.009 and we can conclude that the null hypothesis of equal treatment means is rejected.

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Graded Exercise Set 1 - the morley data (one-way ANOVA with one factor)

data('morley')
head(morley)
with(morley, tapply(Speed, Expt, length))
##  1  2  3  4  5 
## 20 20 20 20 20

1. Perform a one-way ANOVA to find out whether the five experiments give different average results for the measured speed of light. Give a p value to test the null hypothesis that the five experiments give the same mean speed of light measurement. What is the decision? What would be a good plot to visualize results?

morley$Expt = as.factor(morley$Expt)
morley.lm = lm(Speed~Expt, data=morley)
library(car)
Anova(morley.lm, type=2)

The overall F test is significant (p-value = 0.003114), hence we can reject the null hypothesis of equal means for each experiment.

library(ggplot2)
ggplot(morley, aes(Expt, Speed)) + geom_jitter(width=0.1, height=0)

#or

plot(Speed~Expt, data=morley)

We can see that Experiment 1 has a high variance, and Experiment 3 has three outliers, which could be problematic for the analysis.

2. Which of the five experiments differ from which other ones with respect to the average speed of light?

library(agricolae)
HSD.test(morley.lm, 'Expt', group=TRUE, console=TRUE)
## 
## Study: morley.lm ~ "Expt"
## 
## HSD Test for Speed 
## 
## Mean Square Error:  5510.632 
## 
## Expt,  means
## 
##   Speed       std  r Min  Max
## 1 909.0 104.92604 20 650 1070
## 2 856.0  61.16414 20 760  960
## 3 845.0  79.10686 20 620  970
## 4 820.5  60.04165 20 720  920
## 5 831.5  54.21934 20 740  950
## 
## Alpha: 0.05 ; DF Error: 95 
## Critical Value of Studentized Range: 3.932736 
## 
## Minimun Significant Difference: 65.28006 
## 
## Treatments with the same letter are not significantly different.
## 
##   Speed groups
## 1 909.0      a
## 2 856.0     ab
## 3 845.0     ab
## 5 831.5      b
## 4 820.5      b
#or emmeans 

library(emmeans)
ref_grid(morley.lm)
## 'emmGrid' object with variables:
##     Expt = 1, 2, 3, 4, 5
morley.emm = emmeans(morley.lm, 'Expt')
morley.emm
##  Expt emmean   SE df lower.CL upper.CL
##  1       909 16.6 95      876      942
##  2       856 16.6 95      823      889
##  3       845 16.6 95      812      878
##  4       820 16.6 95      788      853
##  5       832 16.6 95      799      864
## 
## Confidence level used: 0.95
pairs(morley.emm)
##  contrast estimate   SE df t.ratio p.value
##  1 - 2        53.0 23.5 95  2.258  0.1680 
##  1 - 3        64.0 23.5 95  2.726  0.0575 
##  1 - 4        88.5 23.5 95  3.770  0.0026 
##  1 - 5        77.5 23.5 95  3.301  0.0116 
##  2 - 3        11.0 23.5 95  0.469  0.9900 
##  2 - 4        35.5 23.5 95  1.512  0.5572 
##  2 - 5        24.5 23.5 95  1.044  0.8343 
##  3 - 4        24.5 23.5 95  1.044  0.8343 
##  3 - 5        13.5 23.5 95  0.575  0.9784 
##  4 - 5       -11.0 23.5 95 -0.469  0.9900 
## 
## P value adjustment: tukey method for comparing a family of 5 estimates

The output indicates a significant difference in mean light speed between Experiment 1 and 4 and Experiment 1 and 5.

3. Test the normality assumption. Describe the QQ plot in a few words and give the p value of a formal test for normality. What are your conclusions regarding normality?

library(car)
qqPlot(resid(morley.lm))

## 014 047 
##  14  47

We can see that the distribution has a heavy lower tail. The qqPlot shows problems with nonnormality. For further assessment one could perform a Shapiro test:

shapiro.test(resid(morley.lm))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(morley.lm)
## W = 0.96779, p-value = 0.01501

The null hypothesis of normaility is rejected with a p-value of 0.01501.

4. Test for homogeneity of variances. Give p values for the tests and interpret.

bartlett.test(Speed~Expt, data=morley)
## 
##  Bartlett test of homogeneity of variances
## 
## data:  Speed by Expt
## Bartlett's K-squared = 11.552, df = 4, p-value = 0.02102
leveneTest(Speed~Expt, data=morley)
plot(morley.lm, which=1)

The Bartlett test has a p-value of 0.02102 and we would have to reject the null hypothesis of equal variance. The Levene’s Test does not reject the null hypothesis of equal variance, this could be due to the fact that the Bartlett test is very sensitive to non normality. The residual vs. fitted plot also shows some problems with variance, since it increases with the fitted values.

5. Perform a non-parametric test to answer the question whether the five experiments tend to give different results for the distribution of the speed of light.

kruskal.test(Speed~Expt, data=morley)
## 
##  Kruskal-Wallis rank sum test
## 
## data:  Speed by Expt
## Kruskal-Wallis chi-squared = 15.022, df = 4, p-value = 0.004656
#or use a robust method

library(asbio)
with(morley, BDM.test(Speed, Expt))
## 
## One way Brunner-Dette-Munk test 
## 
##       df1      df2       F*   P(F > F*)
##  3.819825 89.37885 4.248446 0.003906571

We perform a Kruskal-Wallis test even though the test suffers if we do not have equal variances. We reject the null hypothesis that the Experiments are coming from the same distribution (p-value = 0.004656). The Brunner Dette Munk test comes to the same conclusion (p-value = 0.0039).

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Local Teaching Session 3 - the winterwheat data (RCBD - mixed effect model)

The yan.winterwheat data frame from the agridat library contains data on the yield (in tons per ha) of 18 genotypes (variable gen, this means varieties) of winter wheat grown in 9 environments (variable env, these are locations). Each variety was randomly allocated to exactly one plot for each location. Our aim to model the effect of the different genotypes and the different environments on the yield.

library(agridat)
data('yan.winterwheat')
head(yan.winterwheat)

1. What are the treatments, how many treatments are there, and how many replications per treatment does this data set provide?

unique(yan.winterwheat$gen)
##  [1] Ann Ari Aug Cas Del Dia Ena Fun Ham Har Kar Kat Luc m12 Reb Ron Rub Zav
## 18 Levels: Ann Ari Aug Cas Del Dia Ena Fun Ham Har Kar Kat Luc Reb Ron ... m12
with(yan.winterwheat, tapply(yield, gen, length))
## Ann Ari Aug Cas Del Dia Ena Fun Ham Har Kar Kat Luc Reb Ron Rub Zav m12 
##   9   9   9   9   9   9   9   9   9   9   9   9   9   9   9   9   9   9
with(yan.winterwheat, table(env, gen))
##       gen
## env    Ann Ari Aug Cas Del Dia Ena Fun Ham Har Kar Kat Luc Reb Ron Rub Zav m12
##   BH93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##   EA93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##   HW93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##   ID93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##   KE93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##   NN93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##   OA93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##   RN93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##   WP93   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
18*9
## [1] 162

We have 18 different treatments which grow once per each of the nine environments. The treatments are the 18$$9 = 162 combinations of genotype and environment. Each treatment is observed exactly one time.

2. The main research interest is in the genotype effect on the average yield, not in the effect of the particular environments used in this study (only in the variability induced by the environments). What does this imply for the statistical model used in the analysis?

Since we are not interested in the fixed effect of the environments, but only in the random effects we should fit a mixed effect model to the data. With genotype being the fixed effect and environment a random effect, since we only observe each treatment once we cannot model the interaction between genotype and environment.

3. Give the R code to fit a suitable parametric model to this data set to answer the research question.

library(nlme)
ww.lme = lme(yield~gen, random=~1|env, data=yan.winterwheat)

4. Is the genotype effect significant according to your model from 3? Name the tools you use, give your R code to perform the tests, report the p value and use it to answer the question.

anova(ww.lme, type='marginal')

I use anova to perform an overall F test, the result is a significant genotype effect (p-value < 0.0001).

5. According to the model in 3, how high is the average yield of the Kat genotype for a “typical” location? Explain your approach.

summary(ww.lme)
## Linear mixed-effects model fit by REML
##  Data: yan.winterwheat 
##        AIC      BIC    logLik
##   248.5094 307.9057 -104.2547
## 
## Random effects:
##  Formula: ~1 | env
##         (Intercept)  Residual
## StdDev:   0.8872641 0.3830808
## 
## Fixed effects: yield ~ gen 
##                 Value Std.Error  DF   t-value p-value
## (Intercept)  3.999444 0.3221436 136 12.415098  0.0000
## genAri       0.304667 0.1805860 136  1.687100  0.0939
## genAug       0.239556 0.1805860 136  1.326545  0.1869
## genCas       0.543889 0.1805860 136  3.011799  0.0031
## genDel       0.403111 0.1805860 136  2.232239  0.0272
## genDia       0.342444 0.1805860 136  1.896295  0.0600
## genEna      -0.297889 0.1805860 136 -1.649568  0.1013
## genFun       0.674222 0.1805860 136  3.733524  0.0003
## genHam       0.433111 0.1805860 136  2.398364  0.0178
## genHar       0.460444 0.1805860 136  2.549723  0.0119
## genKar       0.282667 0.1805860 136  1.565274  0.1198
## genKat      -0.780222 0.1805860 136 -4.320501  0.0000
## genLuc      -0.205667 0.1805860 136 -1.138885  0.2568
## genReb       0.302667 0.1805860 136  1.676025  0.0960
## genRon       0.390222 0.1805860 136  2.160866  0.0325
## genRub       0.236889 0.1805860 136  1.311779  0.1918
## genZav       0.486333 0.1805860 136  2.693084  0.0080
## genm12      -0.457778 0.1805860 136 -2.534957  0.0124
##  Correlation: 
##        (Intr) genAri genAug genCas genDel genDia genEna genFun genHam genHar
## genAri -0.28                                                                
## genAug -0.28   0.50                                                         
## genCas -0.28   0.50   0.50                                                  
## genDel -0.28   0.50   0.50   0.50                                           
## genDia -0.28   0.50   0.50   0.50   0.50                                    
## genEna -0.28   0.50   0.50   0.50   0.50   0.50                             
## genFun -0.28   0.50   0.50   0.50   0.50   0.50   0.50                      
## genHam -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50               
## genHar -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50        
## genKar -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50 
## genKat -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50 
## genLuc -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50 
## genReb -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50 
## genRon -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50 
## genRub -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50 
## genZav -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50 
## genm12 -0.28   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50   0.50 
##        genKar genKat genLuc genReb genRon genRub genZav
## genAri                                                 
## genAug                                                 
## genCas                                                 
## genDel                                                 
## genDia                                                 
## genEna                                                 
## genFun                                                 
## genHam                                                 
## genHar                                                 
## genKar                                                 
## genKat  0.50                                           
## genLuc  0.50   0.50                                    
## genReb  0.50   0.50   0.50                             
## genRon  0.50   0.50   0.50   0.50                      
## genRub  0.50   0.50   0.50   0.50   0.50               
## genZav  0.50   0.50   0.50   0.50   0.50   0.50        
## genm12  0.50   0.50   0.50   0.50   0.50   0.50   0.50 
## 
## Standardized Within-Group Residuals:
##        Min         Q1        Med         Q3        Max 
## -3.2610953 -0.6274208 -0.0567675  0.5017550  2.5838475 
## 
## Number of Observations: 162
## Number of Groups: 9
#or 

fixef(ww.lme)['(Intercept)'] + fixef(ww.lme)['genKat']
## (Intercept) 
##    3.219222

The average yield of the reference level is 3.999444 and the yield of the Kat genotype is on average 0.780222 lower. We get an avergae yield for the Kat genotype of 3.219222 tons per ha.

6. Assess the model assumptions: explain what you assess, with which method, give your R code, discuss the results (give p values where you calculate them) and report your conclusions regarding the assumptions.

# normality
library(car)
qqPlot(resid(ww.lme))

## OA93 RN93 
##  121  139
shapiro.test(resid(ww.lme))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(ww.lme)
## W = 0.99089, p-value = 0.3886
plot(ww.lme)

We test for normality and equal variance of the error terms. Testing for normality of the random effect does not make sense since we only have 9 environments. Both the qqPlot and the shapiro test (p-value = 0.3886) do not show any problems with normality. The equal variance assumption for the error terms seems to be violated, the variance of the residuals seems to increase with the fitted values. A solution for this would be to model heteroscedastic within group errors, with a separate variance for each level of the gen factor.

ww.lme.het = lme(yield~gen, random=~1|env, data=yan.winterwheat, weights=varIdent(form=~1|gen))

plot(ww.lme.het)

7. Regardless of the model diagnostics, name an appropriate nonparametric method to analyze this data set, perform this analysis and give the results and your con- clusion.

friedman.test(yield~gen|env, yan.winterwheat)
## 
##  Friedman rank sum test
## 
## data:  yield and gen and env
## Friedman chi-squared = 75.222, df = 17, p-value = 2.671e-09

The Friedman test rejects the null hypothesis (p-value = 2.671e-09) that the treatment effect is the same for every level of the treatment. Loss of power if the data do come from a normal distribution!

8. A researcher argues: “To construct a confidence interval for the average yield of the Kat variety, we can simply use the classical confidence interval based on the t distribution, using the nine values (from the nine environments) that we have for the Kat genotype.”

The researcher correctly implements this idea as follows:

Kat.yield <- yan.winterwheat$yield[yan.winterwheat$gen == "Kat"]
as.numeric( t.test(Kat.yield, alternative = "two.sided")$conf.int)
## [1] 2.577343 3.861101

The true average yield of the Kat variety is between 2.577 and 3.861 tons per ha with a confidence of 95%

9. Disregarding questions about power or model assumptions, what is the main flaw in the approach in 8?

The approach ignores the random effect which is introduced by the different environments, hence evironment variability is not properly accounted for.

10. Do you know a better way of obtaining a confidence interval for the average of the Kat genotype? If yes, explain your approach, give your code and the resulting interval.

We use emmans:

library(emmeans)
ww.emm = emmeans(ww.lme, 'gen')
confint(ww.emm, adjust='none')

The true yield for a typical environment for the Kat genotyple lies between 2.48 and 3.96 tons/ha with a confidence of 95%.

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Graded Exercise Set 2 - the metal data (RCBD - mixed effect model)

“An experiment was conducted to compare two protective dyes for metal, both with each other and with ‘no dye’. Ten braided metal cords were broken into three pieces. The three pieces of each cord were randomly allocated to the three treatments. (. . . ) After the dyes had been applied, the cords were left to weather for a fixed time, then their strengths were measured.”

The file contains the strengths as a percentage of the nominal strength specification, it also contains the sums per cord and per treatment.

metal = read.csv('metal.csv', sep=',', dec='.', header=TRUE)
metal

1. This special design has a name. How is it called, and what is the defining charac- teristic of this design (argue with how randomization occurs)?

The design is called a Randomized Complete Block design (RCBD) since every treatment (dyes) is allocated exactly one time in every plot and treatments are allocated randomly within blocks.

2. The data is not yet in a suitable form for analysis. Explain how the data should look like for R and explain how you produce a suitable data set. Try to find a solution which scales, i. e. which would also work well if you had a much bigger data set.

The data has to be converted to a tidy long format. First we exclude the row with the sum of all strength measurements per treatment, then we use the melt() function to convert the data frame to the long format. We choose the Cord as the id variable and melt the treatments which where stored as three separate columns into one column corresponding now to the treatment.

library(reshape2)
metal1 = metal[metal$Cord != 'Treatment total',]
mmetal = melt(metal1, id.vars = 'Cord', measure.vars = c("No.dye", "Dye.A", 'Dye.B'), variable.name='treatment', value.name='strength')
mmetal

3. Explain what type of model is appropriate for this research setting and give the R code for fitting this model to the data set you produced above. Perform the required model diagnostics.

We are interested in the fixed effects of the treatments, while accounting for the random effects of the blocks. We can fit a mixed effect model to the data. We assess the model assumptions of normality of the residuals and the equal variance assumption of the residuals. We do not assess the normality of the random intercepts since we only have 10 blocks.

metal.lme = lme(strength~treatment, random=~1|Cord, data=mmetal)

library(car)
qqPlot(resid(metal.lme))

##  5  9 
## 15 29
shapiro.test(resid(metal.lme))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(metal.lme)
## W = 0.98163, p-value = 0.867
plot(metal.lme)

No problems with normality or equal variance can be detected.

4. What standard deviations are estimated, and how big are their values? Comment on the results.

summary(metal.lme)
## Linear mixed-effects model fit by REML
##  Data: mmetal 
##        AIC      BIC    logLik
##   181.5369 188.0161 -85.76846
## 
## Random effects:
##  Formula: ~1 | Cord
##         (Intercept) Residual
## StdDev:    4.333235  3.95704
## 
## Fixed effects: strength ~ treatment 
##                Value Std.Error DF  t-value p-value
## (Intercept)    96.67  1.855669 18 52.09441  0.0000
## treatmentDye.A  2.62  1.769642 18  1.48053  0.1560
## treatmentDye.B  4.95  1.769642 18  2.79718  0.0119
##  Correlation: 
##                (Intr) trtD.A
## treatmentDye.A -0.477       
## treatmentDye.B -0.477  0.500
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -1.85678766 -0.57511066  0.08508085  0.38943995  1.54081610 
## 
## Number of Observations: 30
## Number of Groups: 10

The standard deviation of the random intercept of the cord is given with 4.333235, while the standard deviation of the residuals is given with 3.95704. The importance of the blocking variable is reflected in the large standard deviation of the random intercept.

5. According to the model, how high are the average strength percentages per treatment? Do you find a significant treatment effect? Which of the comparisons stated as the aim of the experiment are significant?

#first overall F test
anova(metal.lme, type='marginal')
#pairwise comparisons
library(emmeans)
metal.emm = emmeans(metal.lme, 'treatment')
metal.emm
##  treatment emmean   SE df lower.CL upper.CL
##  No.dye      96.7 1.86  9     92.5      101
##  Dye.A       99.3 1.86  9     95.1      103
##  Dye.B      101.6 1.86  9     97.4      106
## 
## Degrees-of-freedom method: containment 
## Confidence level used: 0.95
pairs(metal.emm)
##  contrast       estimate   SE df t.ratio p.value
##  No.dye - Dye.A    -2.62 1.77 18 -1.481  0.3233 
##  No.dye - Dye.B    -4.95 1.77 18 -2.797  0.0305 
##  Dye.A - Dye.B     -2.33 1.77 18 -1.317  0.4044 
## 
## Degrees-of-freedom method: containment 
## P value adjustment: tukey method for comparing a family of 3 estimates

The overall F test finds a significant treatment effect. The average strength percentage of the control (no dye) is 96.7. The average strength percentage of Dye A is 99.3 and of Dye B 101.6. Only the difference of the control treatment and Dye B is significant (p-value = 0.0305). The p-value is adjusted to a familywise error rate of 5%.

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Local Teaching Session 4 - the paper data (multifactor experiment)

paper = read.csv('paper.csv', sep=',', dec='.', header=TRUE)
head(paper)
str(paper)
## 'data.frame':    36 obs. of  4 variables:
##  $ Replicate  : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ Pulp       : int  1 1 1 1 2 2 2 2 3 3 ...
##  $ Temperature: int  200 225 250 275 200 225 250 275 200 225 ...
##  $ Strength   : int  30 35 37 36 34 41 38 42 29 26 ...
paper$Replicate = factor(paper$Replicate)
paper$Pulp = factor(paper$Pulp)
paper$fTemperature = factor(paper$Temperature)
str(paper)
## 'data.frame':    36 obs. of  5 variables:
##  $ Replicate   : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Pulp        : Factor w/ 3 levels "1","2","3": 1 1 1 1 2 2 2 2 3 3 ...
##  $ Temperature : int  200 225 250 275 200 225 250 275 200 225 ...
##  $ Strength    : int  30 35 37 36 34 41 38 42 29 26 ...
##  $ fTemperature: Factor w/ 4 levels "200","225","250",..: 1 2 3 4 1 2 3 4 1 2 ...
library(ggplot2)
ggplot(paper, aes(x=Temperature, y=Strength, col=Pulp)) + geom_point() + geom_line() +  facet_wrap(~Replicate)

In the second Replicate for Temperature 275 and Pulp 1 there is an outlier, which is actually not a valid data point if we compare the point to the results in the paper.

paper$Strength[(paper$Replicate == '2') & (paper$Pulp) == '1' & (paper$Temperature) == '275'] = 41
paper$fTemperature = factor(paper$Temperature)

library(ggplot2)
ggplot(paper, aes(x=Temperature, y=Strength, col=Pulp)) + geom_point() + geom_line() + facet_wrap(~Replicate)

with(paper, tapply(Strength, list(Pulp, Temperature), mean))
##        200      225      250      275
## 1 29.66667 34.66667 39.33333 39.00000
## 2 33.33333 39.00000 39.66667 42.00000
## 3 30.66667 30.00000 34.66667 40.33333
with(paper, tapply(Strength, list(Pulp, Temperature), sd))
##        200      225      250      275
## 1 1.527525 2.516611 2.081666 2.645751
## 2 2.081666 2.645751 2.081666 2.000000
## 3 1.527525 4.000000 3.785939 4.509250
library(nlme)

paper.lme = lme(Strength ~ fTemperature*Pulp, random=~1|Replicate/Pulp, data=paper)

library(car)
qqPlot(resid(paper.lme))

## 2/2 2/1 
##  19  16
plot(paper.lme)

No problems with the normality of the residuals or with the euqal vairance assumption.

summary(paper.lme)
## Linear mixed-effects model fit by REML
##  Data: paper 
##        AIC      BIC   logLik
##   152.2556 169.9264 -61.1278
## 
## Random effects:
##  Formula: ~1 | Replicate
##         (Intercept)
## StdDev:    1.573434
## 
##  Formula: ~1 | Pulp %in% Replicate
##         (Intercept) Residual
## StdDev:    1.128851 1.993044
## 
## Fixed effects: Strength ~ fTemperature * Pulp 
##                           Value Std.Error DF   t-value p-value
## (Intercept)           29.666667  1.604392 18 18.490908  0.0000
## fTemperature225        5.000000  1.627313 18  3.072549  0.0066
## fTemperature250        9.666667  1.627313 18  5.940262  0.0000
## fTemperature275        9.333333  1.627313 18  5.735425  0.0000
## Pulp2                  3.666667  1.870210  4  1.960564  0.1215
## Pulp3                  1.000000  1.870210  4  0.534699  0.6212
## fTemperature225:Pulp2  0.666667  2.301369 18  0.289683  0.7754
## fTemperature250:Pulp2 -3.333333  2.301369 18 -1.448414  0.1647
## fTemperature275:Pulp2 -0.666667  2.301369 18 -0.289683  0.7754
## fTemperature225:Pulp3 -5.666667  2.301369 18 -2.462303  0.0241
## fTemperature250:Pulp3 -5.666667  2.301369 18 -2.462303  0.0241
## fTemperature275:Pulp3  0.333333  2.301369 18  0.144841  0.8864
##  Correlation: 
##                       (Intr) fTm225 fTm250 fTm275 Pulp2  Pulp3  fT225:P2
## fTemperature225       -0.507                                            
## fTemperature250       -0.507  0.500                                     
## fTemperature275       -0.507  0.500  0.500                              
## Pulp2                 -0.583  0.435  0.435  0.435                       
## Pulp3                 -0.583  0.435  0.435  0.435  0.500                
## fTemperature225:Pulp2  0.359 -0.707 -0.354 -0.354 -0.615 -0.308         
## fTemperature250:Pulp2  0.359 -0.354 -0.707 -0.354 -0.615 -0.308  0.500  
## fTemperature275:Pulp2  0.359 -0.354 -0.354 -0.707 -0.615 -0.308  0.500  
## fTemperature225:Pulp3  0.359 -0.707 -0.354 -0.354 -0.308 -0.615  0.500  
## fTemperature250:Pulp3  0.359 -0.354 -0.707 -0.354 -0.308 -0.615  0.250  
## fTemperature275:Pulp3  0.359 -0.354 -0.354 -0.707 -0.308 -0.615  0.250  
##                       fT250:P2 fT275:P2 fT225:P3 fT250:P3
## fTemperature225                                          
## fTemperature250                                          
## fTemperature275                                          
## Pulp2                                                    
## Pulp3                                                    
## fTemperature225:Pulp2                                    
## fTemperature250:Pulp2                                    
## fTemperature275:Pulp2  0.500                             
## fTemperature225:Pulp3  0.250    0.250                    
## fTemperature250:Pulp3  0.500    0.250    0.500           
## fTemperature275:Pulp3  0.250    0.500    0.500    0.500  
## 
## Standardized Within-Group Residuals:
##        Min         Q1        Med         Q3        Max 
## -1.1366311 -0.6873645  0.1481579  0.5103748  1.6541725 
## 
## Number of Observations: 36
## Number of Groups: 
##           Replicate Pulp %in% Replicate 
##                   3                   9
anova(paper.lme, type='marginal')

The interaction effect of the temperature and the pulp is significant (p-value = 0.0271), hence we cannot simplify the model further.

We could exploit the ordered structure of the temperature factor:

paper$oTemperature = factor(paper$Temperature, ordered=TRUE)

paper.lme.o = lme(Strength~oTemperature*Pulp, random=~1|Replicate/Pulp, data=paper)

coef(summary(paper.lme.o))
##                           Value Std.Error DF     t-value      p-value
## (Intercept)          35.6666667  1.257385 18 28.36573760 2.154157e-16
## oTemperature.L        7.3044887  1.150684 18  6.34795219 5.570280e-06
## oTemperature.Q       -2.6666667  1.150684 18 -2.31746165 3.245900e-02
## oTemperature.C       -1.0434984  1.150684 18 -0.90685031 3.764665e-01
## Pulp2                 2.8333333  1.229460  4  2.30453402 8.252652e-02
## Pulp3                -1.7500000  1.229460  4 -1.42338866 2.277160e-01
## oTemperature.L:Pulp2 -1.3416408  1.627313 18 -0.82445144 4.204745e-01
## oTemperature.Q:Pulp2  1.0000000  1.627313 18  0.61450982 5.465684e-01
## oTemperature.C:Pulp2  2.5342104  1.627313 18  1.55729716 1.368077e-01
## oTemperature.L:Pulp3  0.2236068  1.627313 18  0.13740857 8.922332e-01
## oTemperature.Q:Pulp3  5.8333333  1.627313 18  3.58464061 2.118345e-03
## oTemperature.C:Pulp3  0.0745356  1.627313 18  0.04580286 9.639717e-01
anova(paper.lme.o, type='marginal')

Only the cubic trend is not significant. The interaction effect of the pulp and treatment is still significant (p-value = 0.0271).

coef(summary(paper.lme))
##                            Value Std.Error DF    t-value      p-value
## (Intercept)           29.6666667  1.604392 18 18.4909081 3.714897e-13
## fTemperature225        5.0000000  1.627313 18  3.0725491 6.561433e-03
## fTemperature250        9.6666667  1.627313 18  5.9402616 1.273963e-05
## fTemperature275        9.3333333  1.627313 18  5.7354250 1.946204e-05
## Pulp2                  3.6666667  1.870210  4  1.9605645 1.214760e-01
## Pulp3                  1.0000000  1.870210  4  0.5346994 6.211966e-01
## fTemperature225:Pulp2  0.6666667  2.301369 18  0.2896827 7.753694e-01
## fTemperature250:Pulp2 -3.3333333  2.301369 18 -1.4484135 1.646977e-01
## fTemperature275:Pulp2 -0.6666667  2.301369 18 -0.2896827 7.753694e-01
## fTemperature225:Pulp3 -5.6666667  2.301369 18 -2.4623030 2.412202e-02
## fTemperature250:Pulp3 -5.6666667  2.301369 18 -2.4623030 2.412202e-02
## fTemperature275:Pulp3  0.3333333  2.301369 18  0.1448414 8.864456e-01
pred.df = expand.grid(fTemperature=levels(paper$fTemperature), Pulp=levels(paper$Pulp))
pred.df$fitted = predict(paper.lme, pred.df, level=0)
pred.df
ggplot(pred.df, aes(x=fTemperature, y=fitted, col=Pulp)) + geom_point() + geom_line(aes(group=Pulp))

library(emmeans)

paper.emm = emmeans(paper.lme, pairwise~Pulp|fTemperature)

pairs(paper.emm)
## $emmeans
## fTemperature = 200:
##  contrast estimate   SE df t.ratio p.value
##  1 - 2      -3.667 1.87  4 -1.961  0.2371 
##  1 - 3      -1.000 1.87  4 -0.535  0.8593 
##  2 - 3       2.667 1.87  4  1.426  0.4119 
## 
## fTemperature = 225:
##  contrast estimate   SE df t.ratio p.value
##  1 - 2      -4.333 1.87  4 -2.317  0.1639 
##  1 - 3       4.667 1.87  4  2.495  0.1367 
##  2 - 3       9.000 1.87  4  4.812  0.0186 
## 
## fTemperature = 250:
##  contrast estimate   SE df t.ratio p.value
##  1 - 2      -0.333 1.87  4 -0.178  0.9827 
##  1 - 3       4.667 1.87  4  2.495  0.1367 
##  2 - 3       5.000 1.87  4  2.673  0.1144 
## 
## fTemperature = 275:
##  contrast estimate   SE df t.ratio p.value
##  1 - 2      -3.000 1.87  4 -1.604  0.3436 
##  1 - 3      -1.333 1.87  4 -0.713  0.7696 
##  2 - 3       1.667 1.87  4  0.891  0.6735 
## 
## Degrees-of-freedom method: containment 
## P value adjustment: tukey method for comparing a family of 3 estimates 
## 
## $contrasts
## fTemperature = 200:
##  contrast          estimate   SE df t.ratio p.value
##  (1 - 2) - (1 - 3)   -2.667 1.87  4 -1.426  0.4119 
##  (1 - 2) - (2 - 3)   -6.333 3.24  4 -1.955  0.2385 
##  (1 - 3) - (2 - 3)   -3.667 1.87  4 -1.961  0.2371 
## 
## fTemperature = 225:
##  contrast          estimate   SE df t.ratio p.value
##  (1 - 2) - (1 - 3)   -9.000 1.87  4 -4.812  0.0186 
##  (1 - 2) - (2 - 3)  -13.333 3.24  4 -4.116  0.0316 
##  (1 - 3) - (2 - 3)   -4.333 1.87  4 -2.317  0.1639 
## 
## fTemperature = 250:
##  contrast          estimate   SE df t.ratio p.value
##  (1 - 2) - (1 - 3)   -5.000 1.87  4 -2.673  0.1144 
##  (1 - 2) - (2 - 3)   -5.333 3.24  4 -1.646  0.3289 
##  (1 - 3) - (2 - 3)   -0.333 1.87  4 -0.178  0.9827 
## 
## fTemperature = 275:
##  contrast          estimate   SE df t.ratio p.value
##  (1 - 2) - (1 - 3)   -1.667 1.87  4 -0.891  0.6735 
##  (1 - 2) - (2 - 3)   -4.667 3.24  4 -1.441  0.4058 
##  (1 - 3) - (2 - 3)   -3.000 1.87  4 -1.604  0.3436 
## 
## Degrees-of-freedom method: containment 
## P value adjustment: tukey method for comparing a family of 3 estimates

The predicted values show that the fitted values for the strength are the highest for pulp 2 for every temperature, we do not know if the differences are significant yet. We get the highest fitted value for pulp 2 and a temperature of 275, we use emmeans to test if the difference is significant. We do not get a significant difference for any of the pulps at 275 degrees, the only significant difference is between pulp 2 and three at 225 degrees (p-value = 0.0186).

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MOCK EXAM – Version A, Problem 1 (mixed effect model)

mydata1 <- readRDS("tobacco_data.rds")
str(mydata1)
## 'data.frame':    56 obs. of  3 variables:
##  $ block : Factor w/ 8 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 1 2 ...
##  $ dose  : Factor w/ 7 levels "0","250","500",..: 6 2 1 6 6 7 2 4 7 5 ...
##  $ height: num  1299 1369 1170 1219 1120 ...

1. Give the R code to produce suitable descriptive statistics to describe the data set.

summary(mydata1)
##      block      dose       height      
##  1      : 7   0   :8   Min.   : 607.6  
##  2      : 7   250 :8   1st Qu.: 892.7  
##  3      : 7   500 :8   Median : 999.7  
##  4      : 7   1000:8   Mean   :1012.5  
##  5      : 7   1500:8   3rd Qu.:1126.7  
##  6      : 7   2500:8   Max.   :1421.1  
##  (Other):14   5000:8
table(mydata1$dose, mydata1$height)
##       
##        607.6 627.6 667.8 671.9 776.4 787.1 797.8 827 844.2 846.5 852.4 853.6
##   0        0     0     0     0     0     0     0   0     0     0     0     0
##   250      0     0     0     0     0     1     0   0     0     1     0     0
##   500      0     0     0     0     0     0     0   0     0     0     0     1
##   1000     0     0     1     1     0     0     0   0     0     0     0     0
##   1500     0     1     0     0     0     0     0   0     1     0     0     0
##   2500     0     0     0     0     1     0     0   0     0     0     0     0
##   5000     1     0     0     0     0     0     1   1     0     0     1     0
##       
##        873.4 875.9 898.3 917.9 947.4 947.6 960.4 960.7 968.7 971.7 972.2 975.5
##   0        0     0     1     1     0     0     0     0     0     1     0     0
##   250      0     0     0     0     0     0     0     0     0     0     0     0
##   500      1     0     0     0     0     0     1     0     0     0     1     0
##   1000     0     0     0     0     0     0     0     1     0     0     0     0
##   1500     0     0     0     0     1     0     0     0     0     0     0     0
##   2500     0     0     0     0     0     0     0     0     1     0     0     0
##   5000     0     1     0     0     0     1     0     0     0     0     0     1
##       
##        975.8 990.2 993.8 999.4 1000 1003.3 1004 1006.2 1021.9 1031.5 1031.9
##   0        0     0     0     0    0      0    0      0      0      0      0
##   250      0     0     0     0    0      0    0      0      0      0      1
##   500      0     0     0     0    0      0    1      0      0      0      0
##   1000     0     0     0     0    1      0    0      1      0      0      0
##   1500     1     1     1     1    0      0    0      0      0      0      0
##   2500     0     0     0     0    0      1    0      0      1      0      0
##   5000     0     0     0     0    0      0    0      0      0      1      0
##       
##        1069.7 1076.4 1083.7 1087.1 1093.3 1099.6 1120 1146.9 1169.5 1169.6
##   0         0      0      1      1      0      0    0      0      1      0
##   250       1      1      0      0      0      0    0      1      0      0
##   500       0      0      0      0      0      0    0      0      0      0
##   1000      0      0      0      0      0      1    0      0      0      1
##   1500      0      0      0      0      0      0    0      0      0      0
##   2500      0      0      0      0      1      0    1      0      0      0
##   5000      0      0      0      0      0      0    0      0      0      0
##       
##        1172.6 1173.2 1174.9 1181.3 1219.1 1299.2 1322.4 1343.3 1369.2 1418.9
##   0         1      1      0      0      0      0      0      0      0      0
##   250       0      0      0      0      0      0      0      1      1      0
##   500       0      0      1      1      0      0      1      0      0      0
##   1000      0      0      0      0      0      0      0      0      0      1
##   1500      0      0      0      0      0      0      0      0      0      0
##   2500      0      0      0      0      1      1      0      0      0      0
##   5000      0      0      0      0      0      0      0      0      0      0
##       
##        1421.1
##   0         0
##   250       0
##   500       0
##   1000      0
##   1500      1
##   2500      0
##   5000      0
with(mydata1, table(block, dose))
##      dose
## block 0 250 500 1000 1500 2500 5000
##     1 1   1   1    1    1    1    1
##     2 1   1   1    1    1    1    1
##     3 1   1   1    1    1    1    1
##     4 1   1   1    1    1    1    1
##     5 1   1   1    1    1    1    1
##     6 1   1   1    1    1    1    1
##     7 1   1   1    1    1    1    1
##     8 1   1   1    1    1    1    1
with(mydata1, tapply(height, dose, mean))
##         0       250       500      1000      1500      2500      5000 
## 1059.2500 1083.8750 1042.7750  999.3375  974.9375 1062.7375  864.4125
with(mydata1, tapply(height, dose, sd))
##        0      250      500     1000     1500     2500     5000 
## 115.2085 207.0419 165.8663 248.9255 219.9814 160.6280 130.5180

2. Give the R code to produce suitable graphical representations of the data set. What do you observe? (2)

library(ggplot2)
ggplot(mydata1, aes(dose, height)) + geom_point() + facet_grid(~block)

The spread of the data is very similar over the blocks, we cannot see a block effect.

3. Give the R code to fit a suitable parametric model to this data set. Perform an overall F-test, report its p-value and state your conclusion. (2)

mydata1.lme = lme(height~dose, random=~1 | block, data=mydata1)

anova(mydata1.lme, type='marginal')

We fit a mixed effect model to the data, with a fixed effect for the dose and a random effect for the block. The overall F test gives a non significant treatment effect (dose), the p-value is given with 0.199.

4. 4. Assess the model assumptions for your final model: explain what you assess, with which method, give your R code, discuss the results and state your conclusions. (3)

library(car)
qqPlot(resid(mydata1.lme))

##  1  2 
## 49 50
plot(mydata1.lme)

Both the visual test for normality and the visual test for equal variance do not show any problems, we assume that those model assumptions hold.

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MOCK EXAM – Version A, Problem 2 (two factors, no blocks)

cotton = readRDS('cotton_data.rds')
str(cotton)
## 'data.frame':    144 obs. of  6 variables:
##  $ yield   : num  0.99 1.34 1.26 1.44 1.4 1.36 1.23 1.28 1.56 1.64 ...
##  $ year    : Factor w/ 2 levels "Y1","Y2": 1 1 1 1 1 1 1 1 1 1 ...
##  $ nitrogen: Factor w/ 2 levels "N0","N1": 1 1 1 1 1 1 1 1 1 1 ...
##  $ date    : Factor w/ 4 levels "D1","D2","D3",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ water   : Factor w/ 3 levels "I1","I2","I3": 1 1 1 2 2 2 3 3 3 1 ...
##  $ spacing : Factor w/ 3 levels "S1","S2","S3": 1 2 3 1 2 3 1 2 3 1 ...
c = cotton[cotton$year=='Y1', ]
str(c)
## 'data.frame':    72 obs. of  6 variables:
##  $ yield   : num  0.99 1.34 1.26 1.44 1.4 1.36 1.23 1.28 1.56 1.64 ...
##  $ year    : Factor w/ 2 levels "Y1","Y2": 1 1 1 1 1 1 1 1 1 1 ...
##  $ nitrogen: Factor w/ 2 levels "N0","N1": 1 1 1 1 1 1 1 1 1 1 ...
##  $ date    : Factor w/ 4 levels "D1","D2","D3",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ water   : Factor w/ 3 levels "I1","I2","I3": 1 1 1 2 2 2 3 3 3 1 ...
##  $ spacing : Factor w/ 3 levels "S1","S2","S3": 1 2 3 1 2 3 1 2 3 1 ...
with(c, tapply(yield, list(date, nitrogen), mean))
##          N0       N1
## D1 1.317778 2.773333
## D2 1.888889 3.074444
## D3 1.797778 2.664444
## D4 1.427778 1.735556

D2 has the highest average yield, with 3.074444 units.

cotton.lm = lm(yield~date*nitrogen, data=cotton)
Anova(cotton.lm, type=2)

The interaction effect of date and nitrogen is significant (p-value = 0.002), this means that the nitrogen effect depends on the sowing date.

cotton.lm3 = lm(yield~date*nitrogen*year, data=cotton)
Anova(cotton.lm3, type=2)

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Additional exercise - the soy bean data (multifactorial experiment with and without blocks)

a.) Overall, one genotype clearly produces the highest average yield. Which one, and how high is its average yield?

soy = read.delim('soybean.csv', header=TRUE, sep=';', dec='.')
str(soy)
## 'data.frame':    108 obs. of  4 variables:
##  $ gen  : chr  "Tracy" "Centennial" "N72-137" "N72-3058" ...
##  $ loc  : chr  "Plymouth" "Plymouth" "Plymouth" "Plymouth" ...
##  $ block: chr  "B1" "B1" "B1" "B1" ...
##  $ yield: int  1307 1425 1289 1250 1546 1344 1280 1583 1656 1398 ...
soy$gen = factor(soy$gen)
soy$loc = factor(soy$loc)
soy$block = factor(soy$block)

str(soy)
## 'data.frame':    108 obs. of  4 variables:
##  $ gen  : Factor w/ 12 levels "Centennial","D74-7741",..: 12 1 3 4 5 10 2 7 8 9 ...
##  $ loc  : Factor w/ 3 levels "Clayton","Clinton",..: 3 3 3 3 3 3 3 3 3 3 ...
##  $ block: Factor w/ 3 levels "B1","B2","B3": 1 1 1 1 1 1 1 1 1 1 ...
##  $ yield: int  1307 1425 1289 1250 1546 1344 1280 1583 1656 1398 ...
with(soy, tapply(yield, gen, mean))
## Centennial   D74-7741    N72-137   N72-3058   N72-3148   N73-1102    N73-693 
##   1394.778   1406.444   1483.889   1330.667   1566.000   1501.444   1436.222 
##    N73-877    N73-882     R73-81     R75-12      Tracy 
##   1403.333   1396.667   1373.778   1175.889   1369.889

The genotype N72-3148 produces the highest average yield with 1566.000.

b.) Do you find evidence of an effect of the location, the genotype or their interactionon the average yield? First, give only your R code to answer this question, . . .

soy.lm = lm(yield~gen*loc, data=soy)
Anova(soy.lm, type=2)

c.) . . . then name the tools you use, give p values for each effect and answer the question.

We use an overall F test, we find a significant effect of the genotype (p-value = 0.0004) and for the location (p-value < 0.00001). We do not find a significant interaction effect of genotype and location (p-value = 0.22).

d.) A researcher claims that some genotypes yield more in one location, some in another location. Comment on this in view of c.). Justify your answer.

This could be the case for a single observation but we can say that the average difference in yield is not significant.

e.) Regardless of your result in c.), fit a model without interaction effect and then pairwisely compare all the genotypes based on this model. Give your R code to do all of this and report all genotypes which are significantly different from genotype N72-3058.

soy.lm.n = lm(yield~gen+loc, data=soy)

library(emmeans)
soy.emm = emmeans(soy.lm.n, pairwise~gen)
pairs(soy.emm)
## $emmeans
##  contrast                estimate   SE df t.ratio p.value
##  Centennial - (D74-7741)   -11.67 67.1 94 -0.174  1.0000 
##  Centennial - (N72-137)    -89.11 67.1 94 -1.329  0.9732 
##  Centennial - (N72-3058)    64.11 67.1 94  0.956  0.9982 
##  Centennial - (N72-3148)  -171.22 67.1 94 -2.553  0.3216 
##  Centennial - (N73-1102)  -106.67 67.1 94 -1.590  0.9083 
##  Centennial - (N73-693)    -41.44 67.1 94 -0.618  1.0000 
##  Centennial - (N73-877)     -8.56 67.1 94 -0.128  1.0000 
##  Centennial - (N73-882)     -1.89 67.1 94 -0.028  1.0000 
##  Centennial - (R73-81)      21.00 67.1 94  0.313  1.0000 
##  Centennial - (R75-12)     218.89 67.1 94  3.264  0.0638 
##  Centennial - Tracy         24.89 67.1 94  0.371  1.0000 
##  (D74-7741) - (N72-137)    -77.44 67.1 94 -1.155  0.9910 
##  (D74-7741) - (N72-3058)    75.78 67.1 94  1.130  0.9925 
##  (D74-7741) - (N72-3148)  -159.56 67.1 94 -2.379  0.4304 
##  (D74-7741) - (N73-1102)   -95.00 67.1 94 -1.416  0.9574 
##  (D74-7741) - (N73-693)    -29.78 67.1 94 -0.444  1.0000 
##  (D74-7741) - (N73-877)      3.11 67.1 94  0.046  1.0000 
##  (D74-7741) - (N73-882)      9.78 67.1 94  0.146  1.0000 
##  (D74-7741) - (R73-81)      32.67 67.1 94  0.487  1.0000 
##  (D74-7741) - (R75-12)     230.56 67.1 94  3.437  0.0392 
##  (D74-7741) - Tracy         36.56 67.1 94  0.545  1.0000 
##  (N72-137) - (N72-3058)    153.22 67.1 94  2.284  0.4947 
##  (N72-137) - (N72-3148)    -82.11 67.1 94 -1.224  0.9857 
##  (N72-137) - (N73-1102)    -17.56 67.1 94 -0.262  1.0000 
##  (N72-137) - (N73-693)      47.67 67.1 94  0.711  0.9999 
##  (N72-137) - (N73-877)      80.56 67.1 94  1.201  0.9877 
##  (N72-137) - (N73-882)      87.22 67.1 94  1.300  0.9771 
##  (N72-137) - (R73-81)      110.11 67.1 94  1.642  0.8888 
##  (N72-137) - (R75-12)      308.00 67.1 94  4.592  0.0008 
##  (N72-137) - Tracy         114.00 67.1 94  1.700  0.8639 
##  (N72-3058) - (N72-3148)  -235.33 67.1 94 -3.509  0.0318 
##  (N72-3058) - (N73-1102)  -170.78 67.1 94 -2.546  0.3254 
##  (N72-3058) - (N73-693)   -105.56 67.1 94 -1.574  0.9141 
##  (N72-3058) - (N73-877)    -72.67 67.1 94 -1.083  0.9947 
##  (N72-3058) - (N73-882)    -66.00 67.1 94 -0.984  0.9977 
##  (N72-3058) - (R73-81)     -43.11 67.1 94 -0.643  1.0000 
##  (N72-3058) - (R75-12)     154.78 67.1 94  2.308  0.4787 
##  (N72-3058) - Tracy        -39.22 67.1 94 -0.585  1.0000 
##  (N72-3148) - (N73-1102)    64.56 67.1 94  0.962  0.9981 
##  (N72-3148) - (N73-693)    129.78 67.1 94  1.935  0.7344 
##  (N72-3148) - (N73-877)    162.67 67.1 94  2.425  0.4000 
##  (N72-3148) - (N73-882)    169.33 67.1 94  2.525  0.3381 
##  (N72-3148) - (R73-81)     192.22 67.1 94  2.866  0.1706 
##  (N72-3148) - (R75-12)     390.11 67.1 94  5.816  <.0001 
##  (N72-3148) - Tracy        196.11 67.1 94  2.924  0.1496 
##  (N73-1102) - (N73-693)     65.22 67.1 94  0.972  0.9979 
##  (N73-1102) - (N73-877)     98.11 67.1 94  1.463  0.9468 
##  (N73-1102) - (N73-882)    104.78 67.1 94  1.562  0.9181 
##  (N73-1102) - (R73-81)     127.67 67.1 94  1.903  0.7541 
##  (N73-1102) - (R75-12)     325.56 67.1 94  4.854  0.0003 
##  (N73-1102) - Tracy        131.56 67.1 94  1.961  0.7175 
##  (N73-693) - (N73-877)      32.89 67.1 94  0.490  1.0000 
##  (N73-693) - (N73-882)      39.56 67.1 94  0.590  1.0000 
##  (N73-693) - (R73-81)       62.44 67.1 94  0.931  0.9986 
##  (N73-693) - (R75-12)      260.33 67.1 94  3.881  0.0099 
##  (N73-693) - Tracy          66.33 67.1 94  0.989  0.9976 
##  (N73-877) - (N73-882)       6.67 67.1 94  0.099  1.0000 
##  (N73-877) - (R73-81)       29.56 67.1 94  0.441  1.0000 
##  (N73-877) - (R75-12)      227.44 67.1 94  3.391  0.0448 
##  (N73-877) - Tracy          33.44 67.1 94  0.499  1.0000 
##  (N73-882) - (R73-81)       22.89 67.1 94  0.341  1.0000 
##  (N73-882) - (R75-12)      220.78 67.1 94  3.292  0.0591 
##  (N73-882) - Tracy          26.78 67.1 94  0.399  1.0000 
##  (R73-81) - (R75-12)       197.89 67.1 94  2.950  0.1407 
##  (R73-81) - Tracy            3.89 67.1 94  0.058  1.0000 
##  (R75-12) - Tracy         -194.00 67.1 94 -2.892  0.1608 
## 
## Results are averaged over the levels of: loc 
## P value adjustment: tukey method for comparing a family of 12 estimates 
## 
## $contrasts
##  contrast                                              estimate    SE df
##  (Centennial - (D74-7741)) - (Centennial - (N72-137))    77.444  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - (N72-3058))  -75.778  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - (N72-3148))  159.556  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - (N73-1102))   95.000  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - (N73-693))    29.778  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - (N73-877))    -3.111  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - (N73-882))    -9.778  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - (R73-81))    -32.667  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - (R75-12))   -230.556  67.1 94
##  (Centennial - (D74-7741)) - (Centennial - Tracy)       -36.556  67.1 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (N72-137))    65.778 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (N72-3058))  -87.444 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (N72-3148))  147.889 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (N73-1102))   83.333 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (N73-693))    18.111 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (N73-877))   -14.778 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (N73-882))   -21.444 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (R73-81))    -44.333 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - (R75-12))   -242.222 116.2 94
##  (Centennial - (D74-7741)) - ((D74-7741) - Tracy)       -48.222 116.2 94
##  (Centennial - (D74-7741)) - ((N72-137) - (N72-3058))  -164.889  94.9 94
##  (Centennial - (D74-7741)) - ((N72-137) - (N72-3148))    70.444  94.9 94
##  (Centennial - (D74-7741)) - ((N72-137) - (N73-1102))     5.889  94.9 94
##  (Centennial - (D74-7741)) - ((N72-137) - (N73-693))    -59.333  94.9 94
##  (Centennial - (D74-7741)) - ((N72-137) - (N73-877))    -92.222  94.9 94
##  (Centennial - (D74-7741)) - ((N72-137) - (N73-882))    -98.889  94.9 94
##  (Centennial - (D74-7741)) - ((N72-137) - (R73-81))    -121.778  94.9 94
##  (Centennial - (D74-7741)) - ((N72-137) - (R75-12))    -319.667  94.9 94
##  (Centennial - (D74-7741)) - ((N72-137) - Tracy)       -125.667  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3058) - (N72-3148))  223.667  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3058) - (N73-1102))  159.111  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3058) - (N73-693))    93.889  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3058) - (N73-877))    61.000  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3058) - (N73-882))    54.333  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3058) - (R73-81))     31.444  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3058) - (R75-12))   -166.444  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3058) - Tracy)        27.556  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3148) - (N73-1102))  -76.222  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3148) - (N73-693))  -141.444  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3148) - (N73-877))  -174.333  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3148) - (N73-882))  -181.000  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3148) - (R73-81))   -203.889  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3148) - (R75-12))   -401.778  94.9 94
##  (Centennial - (D74-7741)) - ((N72-3148) - Tracy)      -207.778  94.9 94
##  (Centennial - (D74-7741)) - ((N73-1102) - (N73-693))   -76.889  94.9 94
##  (Centennial - (D74-7741)) - ((N73-1102) - (N73-877))  -109.778  94.9 94
##  (Centennial - (D74-7741)) - ((N73-1102) - (N73-882))  -116.444  94.9 94
##  (Centennial - (D74-7741)) - ((N73-1102) - (R73-81))   -139.333  94.9 94
##  (Centennial - (D74-7741)) - ((N73-1102) - (R75-12))   -337.222  94.9 94
##  (Centennial - (D74-7741)) - ((N73-1102) - Tracy)      -143.222  94.9 94
##  (Centennial - (D74-7741)) - ((N73-693) - (N73-877))    -44.556  94.9 94
##  (Centennial - (D74-7741)) - ((N73-693) - (N73-882))    -51.222  94.9 94
##  (Centennial - (D74-7741)) - ((N73-693) - (R73-81))     -74.111  94.9 94
##  (Centennial - (D74-7741)) - ((N73-693) - (R75-12))    -272.000  94.9 94
##  (Centennial - (D74-7741)) - ((N73-693) - Tracy)        -78.000  94.9 94
##  (Centennial - (D74-7741)) - ((N73-877) - (N73-882))    -18.333  94.9 94
##  (Centennial - (D74-7741)) - ((N73-877) - (R73-81))     -41.222  94.9 94
##  (Centennial - (D74-7741)) - ((N73-877) - (R75-12))    -239.111  94.9 94
##  (Centennial - (D74-7741)) - ((N73-877) - Tracy)        -45.111  94.9 94
##  (Centennial - (D74-7741)) - ((N73-882) - (R73-81))     -34.556  94.9 94
##  (Centennial - (D74-7741)) - ((N73-882) - (R75-12))    -232.444  94.9 94
##  (Centennial - (D74-7741)) - ((N73-882) - Tracy)        -38.444  94.9 94
##  (Centennial - (D74-7741)) - ((R73-81) - (R75-12))     -209.556  94.9 94
##  (Centennial - (D74-7741)) - ((R73-81) - Tracy)         -15.556  94.9 94
##  (Centennial - (D74-7741)) - ((R75-12) - Tracy)         182.333  94.9 94
##  (Centennial - (N72-137)) - (Centennial - (N72-3058))  -153.222  67.1 94
##  (Centennial - (N72-137)) - (Centennial - (N72-3148))    82.111  67.1 94
##  (Centennial - (N72-137)) - (Centennial - (N73-1102))    17.556  67.1 94
##  (Centennial - (N72-137)) - (Centennial - (N73-693))    -47.667  67.1 94
##  (Centennial - (N72-137)) - (Centennial - (N73-877))    -80.556  67.1 94
##  (Centennial - (N72-137)) - (Centennial - (N73-882))    -87.222  67.1 94
##  (Centennial - (N72-137)) - (Centennial - (R73-81))    -110.111  67.1 94
##  (Centennial - (N72-137)) - (Centennial - (R75-12))    -308.000  67.1 94
##  (Centennial - (N72-137)) - (Centennial - Tracy)       -114.000  67.1 94
##  (Centennial - (N72-137)) - ((D74-7741) - (N72-137))    -11.667  67.1 94
##  (Centennial - (N72-137)) - ((D74-7741) - (N72-3058))  -164.889  94.9 94
##  (Centennial - (N72-137)) - ((D74-7741) - (N72-3148))    70.444  94.9 94
##  (Centennial - (N72-137)) - ((D74-7741) - (N73-1102))     5.889  94.9 94
##  (Centennial - (N72-137)) - ((D74-7741) - (N73-693))    -59.333  94.9 94
##  (Centennial - (N72-137)) - ((D74-7741) - (N73-877))    -92.222  94.9 94
##  (Centennial - (N72-137)) - ((D74-7741) - (N73-882))    -98.889  94.9 94
##  (Centennial - (N72-137)) - ((D74-7741) - (R73-81))    -121.778  94.9 94
##  (Centennial - (N72-137)) - ((D74-7741) - (R75-12))    -319.667  94.9 94
##  (Centennial - (N72-137)) - ((D74-7741) - Tracy)       -125.667  94.9 94
##  (Centennial - (N72-137)) - ((N72-137) - (N72-3058))   -242.333 116.2 94
##  (Centennial - (N72-137)) - ((N72-137) - (N72-3148))     -7.000 116.2 94
##  (Centennial - (N72-137)) - ((N72-137) - (N73-1102))    -71.556 116.2 94
##  (Centennial - (N72-137)) - ((N72-137) - (N73-693))    -136.778 116.2 94
##  (Centennial - (N72-137)) - ((N72-137) - (N73-877))    -169.667 116.2 94
##  (Centennial - (N72-137)) - ((N72-137) - (N73-882))    -176.333 116.2 94
##  (Centennial - (N72-137)) - ((N72-137) - (R73-81))     -199.222 116.2 94
##  (Centennial - (N72-137)) - ((N72-137) - (R75-12))     -397.111 116.2 94
##  (Centennial - (N72-137)) - ((N72-137) - Tracy)        -203.111 116.2 94
##  (Centennial - (N72-137)) - ((N72-3058) - (N72-3148))   146.222  94.9 94
##  (Centennial - (N72-137)) - ((N72-3058) - (N73-1102))    81.667  94.9 94
##  (Centennial - (N72-137)) - ((N72-3058) - (N73-693))     16.444  94.9 94
##  (Centennial - (N72-137)) - ((N72-3058) - (N73-877))    -16.444  94.9 94
##  (Centennial - (N72-137)) - ((N72-3058) - (N73-882))    -23.111  94.9 94
##  (Centennial - (N72-137)) - ((N72-3058) - (R73-81))     -46.000  94.9 94
##  (Centennial - (N72-137)) - ((N72-3058) - (R75-12))    -243.889  94.9 94
##  (Centennial - (N72-137)) - ((N72-3058) - Tracy)        -49.889  94.9 94
##  (Centennial - (N72-137)) - ((N72-3148) - (N73-1102))  -153.667  94.9 94
##  (Centennial - (N72-137)) - ((N72-3148) - (N73-693))   -218.889  94.9 94
##  (Centennial - (N72-137)) - ((N72-3148) - (N73-877))   -251.778  94.9 94
##  (Centennial - (N72-137)) - ((N72-3148) - (N73-882))   -258.444  94.9 94
##  (Centennial - (N72-137)) - ((N72-3148) - (R73-81))    -281.333  94.9 94
##  (Centennial - (N72-137)) - ((N72-3148) - (R75-12))    -479.222  94.9 94
##  (Centennial - (N72-137)) - ((N72-3148) - Tracy)       -285.222  94.9 94
##  (Centennial - (N72-137)) - ((N73-1102) - (N73-693))   -154.333  94.9 94
##  (Centennial - (N72-137)) - ((N73-1102) - (N73-877))   -187.222  94.9 94
##  (Centennial - (N72-137)) - ((N73-1102) - (N73-882))   -193.889  94.9 94
##  (Centennial - (N72-137)) - ((N73-1102) - (R73-81))    -216.778  94.9 94
##  (Centennial - (N72-137)) - ((N73-1102) - (R75-12))    -414.667  94.9 94
##  (Centennial - (N72-137)) - ((N73-1102) - Tracy)       -220.667  94.9 94
##  (Centennial - (N72-137)) - ((N73-693) - (N73-877))    -122.000  94.9 94
##  (Centennial - (N72-137)) - ((N73-693) - (N73-882))    -128.667  94.9 94
##  (Centennial - (N72-137)) - ((N73-693) - (R73-81))     -151.556  94.9 94
##  (Centennial - (N72-137)) - ((N73-693) - (R75-12))     -349.444  94.9 94
##  (Centennial - (N72-137)) - ((N73-693) - Tracy)        -155.444  94.9 94
##  (Centennial - (N72-137)) - ((N73-877) - (N73-882))     -95.778  94.9 94
##  (Centennial - (N72-137)) - ((N73-877) - (R73-81))     -118.667  94.9 94
##  (Centennial - (N72-137)) - ((N73-877) - (R75-12))     -316.556  94.9 94
##  (Centennial - (N72-137)) - ((N73-877) - Tracy)        -122.556  94.9 94
##  (Centennial - (N72-137)) - ((N73-882) - (R73-81))     -112.000  94.9 94
##  (Centennial - (N72-137)) - ((N73-882) - (R75-12))     -309.889  94.9 94
##  (Centennial - (N72-137)) - ((N73-882) - Tracy)        -115.889  94.9 94
##  (Centennial - (N72-137)) - ((R73-81) - (R75-12))      -287.000  94.9 94
##  (Centennial - (N72-137)) - ((R73-81) - Tracy)          -93.000  94.9 94
##  (Centennial - (N72-137)) - ((R75-12) - Tracy)          104.889  94.9 94
##  (Centennial - (N72-3058)) - (Centennial - (N72-3148))  235.333  67.1 94
##  (Centennial - (N72-3058)) - (Centennial - (N73-1102))  170.778  67.1 94
##  (Centennial - (N72-3058)) - (Centennial - (N73-693))   105.556  67.1 94
##  (Centennial - (N72-3058)) - (Centennial - (N73-877))    72.667  67.1 94
##  (Centennial - (N72-3058)) - (Centennial - (N73-882))    66.000  67.1 94
##  (Centennial - (N72-3058)) - (Centennial - (R73-81))     43.111  67.1 94
##  (Centennial - (N72-3058)) - (Centennial - (R75-12))   -154.778  67.1 94
##  (Centennial - (N72-3058)) - (Centennial - Tracy)        39.222  67.1 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (N72-137))   141.556  94.9 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (N72-3058))  -11.667  67.1 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (N72-3148))  223.667  94.9 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (N73-1102))  159.111  94.9 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (N73-693))    93.889  94.9 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (N73-877))    61.000  94.9 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (N73-882))    54.333  94.9 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (R73-81))     31.444  94.9 94
##  (Centennial - (N72-3058)) - ((D74-7741) - (R75-12))   -166.444  94.9 94
##  (Centennial - (N72-3058)) - ((D74-7741) - Tracy)        27.556  94.9 94
##  (Centennial - (N72-3058)) - ((N72-137) - (N72-3058))   -89.111  67.1 94
##  (Centennial - (N72-3058)) - ((N72-137) - (N72-3148))   146.222  94.9 94
##  (Centennial - (N72-3058)) - ((N72-137) - (N73-1102))    81.667  94.9 94
##  (Centennial - (N72-3058)) - ((N72-137) - (N73-693))     16.444  94.9 94
##  (Centennial - (N72-3058)) - ((N72-137) - (N73-877))    -16.444  94.9 94
##  (Centennial - (N72-3058)) - ((N72-137) - (N73-882))    -23.111  94.9 94
##  (Centennial - (N72-3058)) - ((N72-137) - (R73-81))     -46.000  94.9 94
##  (Centennial - (N72-3058)) - ((N72-137) - (R75-12))    -243.889  94.9 94
##  (Centennial - (N72-3058)) - ((N72-137) - Tracy)        -49.889  94.9 94
##  (Centennial - (N72-3058)) - ((N72-3058) - (N72-3148))  299.444 116.2 94
##  (Centennial - (N72-3058)) - ((N72-3058) - (N73-1102))  234.889 116.2 94
##  (Centennial - (N72-3058)) - ((N72-3058) - (N73-693))   169.667 116.2 94
##  (Centennial - (N72-3058)) - ((N72-3058) - (N73-877))   136.778 116.2 94
##  (Centennial - (N72-3058)) - ((N72-3058) - (N73-882))   130.111 116.2 94
##  (Centennial - (N72-3058)) - ((N72-3058) - (R73-81))    107.222 116.2 94
##  (Centennial - (N72-3058)) - ((N72-3058) - (R75-12))    -90.667 116.2 94
##  (Centennial - (N72-3058)) - ((N72-3058) - Tracy)       103.333 116.2 94
##  (Centennial - (N72-3058)) - ((N72-3148) - (N73-1102))   -0.444  94.9 94
##  (Centennial - (N72-3058)) - ((N72-3148) - (N73-693))   -65.667  94.9 94
##  (Centennial - (N72-3058)) - ((N72-3148) - (N73-877))   -98.556  94.9 94
##  (Centennial - (N72-3058)) - ((N72-3148) - (N73-882))  -105.222  94.9 94
##  (Centennial - (N72-3058)) - ((N72-3148) - (R73-81))   -128.111  94.9 94
##  (Centennial - (N72-3058)) - ((N72-3148) - (R75-12))   -326.000  94.9 94
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##  (Centennial - (N72-3058)) - ((N73-1102) - (N73-693))    -1.111  94.9 94
##  (Centennial - (N72-3058)) - ((N73-1102) - (N73-877))   -34.000  94.9 94
##  (Centennial - (N72-3058)) - ((N73-1102) - (N73-882))   -40.667  94.9 94
##  (Centennial - (N72-3058)) - ((N73-1102) - (R73-81))    -63.556  94.9 94
##  (Centennial - (N72-3058)) - ((N73-1102) - (R75-12))   -261.444  94.9 94
##  (Centennial - (N72-3058)) - ((N73-1102) - Tracy)       -67.444  94.9 94
##  (Centennial - (N72-3058)) - ((N73-693) - (N73-877))     31.222  94.9 94
##  (Centennial - (N72-3058)) - ((N73-693) - (N73-882))     24.556  94.9 94
##  (Centennial - (N72-3058)) - ((N73-693) - (R73-81))       1.667  94.9 94
##  (Centennial - (N72-3058)) - ((N73-693) - (R75-12))    -196.222  94.9 94
##  (Centennial - (N72-3058)) - ((N73-693) - Tracy)         -2.222  94.9 94
##  (Centennial - (N72-3058)) - ((N73-877) - (N73-882))     57.444  94.9 94
##  (Centennial - (N72-3058)) - ((N73-877) - (R73-81))      34.556  94.9 94
##  (Centennial - (N72-3058)) - ((N73-877) - (R75-12))    -163.333  94.9 94
##  (Centennial - (N72-3058)) - ((N73-877) - Tracy)         30.667  94.9 94
##  (Centennial - (N72-3058)) - ((N73-882) - (R73-81))      41.222  94.9 94
##  (Centennial - (N72-3058)) - ((N73-882) - (R75-12))    -156.667  94.9 94
##  (Centennial - (N72-3058)) - ((N73-882) - Tracy)         37.333  94.9 94
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##  (Centennial - (N72-3058)) - ((R73-81) - Tracy)          60.222  94.9 94
##  (Centennial - (N72-3058)) - ((R75-12) - Tracy)         258.111  94.9 94
##  (Centennial - (N72-3148)) - (Centennial - (N73-1102))  -64.556  67.1 94
##  (Centennial - (N72-3148)) - (Centennial - (N73-693))  -129.778  67.1 94
##  (Centennial - (N72-3148)) - (Centennial - (N73-877))  -162.667  67.1 94
##  (Centennial - (N72-3148)) - (Centennial - (N73-882))  -169.333  67.1 94
##  (Centennial - (N72-3148)) - (Centennial - (R73-81))   -192.222  67.1 94
##  (Centennial - (N72-3148)) - (Centennial - (R75-12))   -390.111  67.1 94
##  (Centennial - (N72-3148)) - (Centennial - Tracy)      -196.111  67.1 94
##  (Centennial - (N72-3148)) - ((D74-7741) - (N72-137))   -93.778  94.9 94
##  (Centennial - (N72-3148)) - ((D74-7741) - (N72-3058)) -247.000  94.9 94
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##  (Centennial - (N72-3148)) - ((D74-7741) - (N73-877))  -174.333  94.9 94
##  (Centennial - (N72-3148)) - ((D74-7741) - (N73-882))  -181.000  94.9 94
##  (Centennial - (N72-3148)) - ((D74-7741) - (R73-81))   -203.889  94.9 94
##  (Centennial - (N72-3148)) - ((D74-7741) - (R75-12))   -401.778  94.9 94
##  (Centennial - (N72-3148)) - ((D74-7741) - Tracy)      -207.778  94.9 94
##  (Centennial - (N72-3148)) - ((N72-137) - (N72-3058))  -324.444  94.9 94
##  (Centennial - (N72-3148)) - ((N72-137) - (N72-3148))   -89.111  67.1 94
##  (Centennial - (N72-3148)) - ((N72-137) - (N73-1102))  -153.667  94.9 94
##  (Centennial - (N72-3148)) - ((N72-137) - (N73-693))   -218.889  94.9 94
##  (Centennial - (N72-3148)) - ((N72-137) - (N73-877))   -251.778  94.9 94
##  (Centennial - (N72-3148)) - ((N72-137) - (N73-882))   -258.444  94.9 94
##  (Centennial - (N72-3148)) - ((N72-137) - (R73-81))    -281.333  94.9 94
##  (Centennial - (N72-3148)) - ((N72-137) - (R75-12))    -479.222  94.9 94
##  (Centennial - (N72-3148)) - ((N72-137) - Tracy)       -285.222  94.9 94
##  (Centennial - (N72-3148)) - ((N72-3058) - (N72-3148))   64.111  67.1 94
##  (Centennial - (N72-3148)) - ((N72-3058) - (N73-1102))   -0.444  94.9 94
##  (Centennial - (N72-3148)) - ((N72-3058) - (N73-693))   -65.667  94.9 94
##  (Centennial - (N72-3148)) - ((N72-3058) - (N73-877))   -98.556  94.9 94
##  (Centennial - (N72-3148)) - ((N72-3058) - (N73-882))  -105.222  94.9 94
##  (Centennial - (N72-3148)) - ((N72-3058) - (R73-81))   -128.111  94.9 94
##  (Centennial - (N72-3148)) - ((N72-3058) - (R75-12))   -326.000  94.9 94
##  (Centennial - (N72-3148)) - ((N72-3058) - Tracy)      -132.000  94.9 94
##  (Centennial - (N72-3148)) - ((N72-3148) - (N73-1102)) -235.778 116.2 94
##  (Centennial - (N72-3148)) - ((N72-3148) - (N73-693))  -301.000 116.2 94
##  (Centennial - (N72-3148)) - ((N72-3148) - (N73-877))  -333.889 116.2 94
##  (Centennial - (N72-3148)) - ((N72-3148) - (N73-882))  -340.556 116.2 94
##  (Centennial - (N72-3148)) - ((N72-3148) - (R73-81))   -363.444 116.2 94
##  (Centennial - (N72-3148)) - ((N72-3148) - (R75-12))   -561.333 116.2 94
##  (Centennial - (N72-3148)) - ((N72-3148) - Tracy)      -367.333 116.2 94
##  (Centennial - (N72-3148)) - ((N73-1102) - (N73-693))  -236.444  94.9 94
##  (Centennial - (N72-3148)) - ((N73-1102) - (N73-877))  -269.333  94.9 94
##  (Centennial - (N72-3148)) - ((N73-1102) - (N73-882))  -276.000  94.9 94
##  (Centennial - (N72-3148)) - ((N73-1102) - (R73-81))   -298.889  94.9 94
##  (Centennial - (N72-3148)) - ((N73-1102) - (R75-12))   -496.778  94.9 94
##  (Centennial - (N72-3148)) - ((N73-1102) - Tracy)      -302.778  94.9 94
##  (Centennial - (N72-3148)) - ((N73-693) - (N73-877))   -204.111  94.9 94
##  (Centennial - (N72-3148)) - ((N73-693) - (N73-882))   -210.778  94.9 94
##  (Centennial - (N72-3148)) - ((N73-693) - (R73-81))    -233.667  94.9 94
##  (Centennial - (N72-3148)) - ((N73-693) - (R75-12))    -431.556  94.9 94
##  (Centennial - (N72-3148)) - ((N73-693) - Tracy)       -237.556  94.9 94
##  (Centennial - (N72-3148)) - ((N73-877) - (N73-882))   -177.889  94.9 94
##  (Centennial - (N72-3148)) - ((N73-877) - (R73-81))    -200.778  94.9 94
##  (Centennial - (N72-3148)) - ((N73-877) - (R75-12))    -398.667  94.9 94
##  (Centennial - (N72-3148)) - ((N73-877) - Tracy)       -204.667  94.9 94
##  (Centennial - (N72-3148)) - ((N73-882) - (R73-81))    -194.111  94.9 94
##  (Centennial - (N72-3148)) - ((N73-882) - (R75-12))    -392.000  94.9 94
##  (Centennial - (N72-3148)) - ((N73-882) - Tracy)       -198.000  94.9 94
##  (Centennial - (N72-3148)) - ((R73-81) - (R75-12))     -369.111  94.9 94
##  (Centennial - (N72-3148)) - ((R73-81) - Tracy)        -175.111  94.9 94
##  (Centennial - (N72-3148)) - ((R75-12) - Tracy)          22.778  94.9 94
##  (Centennial - (N73-1102)) - (Centennial - (N73-693))   -65.222  67.1 94
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##  (Centennial - (N73-1102)) - (Centennial - (R75-12))   -325.556  67.1 94
##  (Centennial - (N73-1102)) - (Centennial - Tracy)      -131.556  67.1 94
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##  (Centennial - (N73-1102)) - ((D74-7741) - (N72-3058)) -182.444  94.9 94
##  (Centennial - (N73-1102)) - ((D74-7741) - (N72-3148))   52.889  94.9 94
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##  (Centennial - (N73-1102)) - ((D74-7741) - (N73-693))   -76.889  94.9 94
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##  (Centennial - (N73-1102)) - ((D74-7741) - (R73-81))   -139.333  94.9 94
##  (Centennial - (N73-1102)) - ((D74-7741) - (R75-12))   -337.222  94.9 94
##  (Centennial - (N73-1102)) - ((D74-7741) - Tracy)      -143.222  94.9 94
##  (Centennial - (N73-1102)) - ((N72-137) - (N72-3058))  -259.889  94.9 94
##  (Centennial - (N73-1102)) - ((N72-137) - (N72-3148))   -24.556  94.9 94
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##  (Centennial - (N73-1102)) - ((N72-137) - (N73-693))   -154.333  94.9 94
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##  (Centennial - (N73-1102)) - ((N72-137) - (R75-12))    -414.667  94.9 94
##  (Centennial - (N73-1102)) - ((N72-137) - Tracy)       -220.667  94.9 94
##  (Centennial - (N73-1102)) - ((N72-3058) - (N72-3148))  128.667  94.9 94
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##  (Centennial - (N73-693)) - (Centennial - (N73-877))    -32.889  67.1 94
##  (Centennial - (N73-693)) - (Centennial - (N73-882))    -39.556  67.1 94
##  (Centennial - (N73-693)) - (Centennial - (R73-81))     -62.444  67.1 94
##  (Centennial - (N73-693)) - (Centennial - (R75-12))    -260.333  67.1 94
##  (Centennial - (N73-693)) - (Centennial - Tracy)        -66.333  67.1 94
##  (Centennial - (N73-693)) - ((D74-7741) - (N72-137))     36.000  94.9 94
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##  (Centennial - (N73-693)) - ((D74-7741) - (N72-3148))   118.111  94.9 94
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##  (Centennial - (N73-693)) - ((D74-7741) - (N73-877))    -44.556  94.9 94
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##  (Centennial - (N73-693)) - ((D74-7741) - (R75-12))    -272.000  94.9 94
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##  (Centennial - (N73-693)) - ((N72-137) - (N72-3148))     40.667  94.9 94
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##  (Centennial - (N73-693)) - ((N72-137) - (N73-693))     -89.111  67.1 94
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##  (Centennial - (N73-693)) - ((N72-137) - (R75-12))     -349.444  94.9 94
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##  (Centennial - (N73-693)) - ((N72-3058) - (N72-3148))   193.889  94.9 94
##  (Centennial - (N73-693)) - ((N72-3058) - (N73-1102))   129.333  94.9 94
##  (Centennial - (N73-693)) - ((N72-3058) - (N73-693))     64.111  67.1 94
##  (Centennial - (N73-693)) - ((N72-3058) - (N73-877))     31.222  94.9 94
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##  (Centennial - (N73-693)) - ((N73-1102) - (N73-877))   -139.556  94.9 94
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##  (Centennial - (N73-693)) - ((N73-693) - (N73-882))     -81.000 116.2 94
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##  (Centennial - (N73-693)) - ((N73-882) - (R75-12))     -262.222  94.9 94
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##  (Centennial - (N73-693)) - ((R75-12) - Tracy)          152.556  94.9 94
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##  (Centennial - (N73-877)) - ((D74-7741) - (N72-3148))   151.000  94.9 94
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##  (Centennial - (N73-877)) - ((N72-137) - (N72-3148))     73.556  94.9 94
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##  (Centennial - (N73-877)) - ((N72-137) - (N73-877))     -89.111  67.1 94
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##  (Centennial - (N73-877)) - ((N72-3058) - (N72-3148))   226.778  94.9 94
##  (Centennial - (N73-877)) - ((N72-3058) - (N73-1102))   162.222  94.9 94
##  (Centennial - (N73-877)) - ((N72-3058) - (N73-693))     97.000  94.9 94
##  (Centennial - (N73-877)) - ((N72-3058) - (N73-877))     64.111  67.1 94
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##  (Centennial - (N73-877)) - ((N72-3148) - (N73-693))   -138.333  94.9 94
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##  (Centennial - (N73-877)) - ((N73-1102) - (N73-693))    -73.778  94.9 94
##  (Centennial - (N73-877)) - ((N73-1102) - (N73-877))   -106.667  67.1 94
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##  (Centennial - (N73-882)) - ((D74-7741) - (N73-693))     27.889  94.9 94
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##  (Centennial - (N73-882)) - ((N72-137) - (N72-3148))     80.222  94.9 94
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##  (Centennial - (N73-882)) - ((N72-137) - (N73-693))     -49.556  94.9 94
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##  (Centennial - (N73-882)) - ((N72-137) - (N73-882))     -89.111  67.1 94
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##  (Centennial - (N73-882)) - ((N72-3058) - (N72-3148))   233.444  94.9 94
##  (Centennial - (N73-882)) - ((N72-3058) - (N73-1102))   168.889  94.9 94
##  (Centennial - (N73-882)) - ((N72-3058) - (N73-693))    103.667  94.9 94
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##  (Centennial - (R73-81)) - ((D74-7741) - (N73-877))      17.889  94.9 94
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##  (Centennial - (R73-81)) - ((D74-7741) - (R75-12))     -209.556  94.9 94
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##  (Centennial - (R73-81)) - ((N72-137) - (N72-3148))     103.111  94.9 94
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##  (Centennial - (R73-81)) - ((N72-137) - (R73-81))       -89.111  67.1 94
##  (Centennial - (R73-81)) - ((N72-137) - (R75-12))      -287.000  94.9 94
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##  (Centennial - (R73-81)) - ((N72-3058) - (N73-693))     126.556  94.9 94
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##  (Centennial - (R73-81)) - ((N72-3058) - (R73-81))       64.111  67.1 94
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##  (Centennial - (R73-81)) - ((N73-1102) - (N73-882))     -83.778  94.9 94
##  (Centennial - (R73-81)) - ((N73-1102) - (R73-81))     -106.667  67.1 94
##  (Centennial - (R73-81)) - ((N73-1102) - (R75-12))     -304.556  94.9 94
##  (Centennial - (R73-81)) - ((N73-1102) - Tracy)        -110.556  94.9 94
##  (Centennial - (R73-81)) - ((N73-693) - (N73-877))      -11.889  94.9 94
##  (Centennial - (R73-81)) - ((N73-693) - (N73-882))      -18.556  94.9 94
##  (Centennial - (R73-81)) - ((N73-693) - (R73-81))       -41.444  67.1 94
##  (Centennial - (R73-81)) - ((N73-693) - (R75-12))      -239.333  94.9 94
##  (Centennial - (R73-81)) - ((N73-693) - Tracy)          -45.333  94.9 94
##  (Centennial - (R73-81)) - ((N73-877) - (N73-882))       14.333  94.9 94
##  (Centennial - (R73-81)) - ((N73-877) - (R73-81))        -8.556  67.1 94
##  (Centennial - (R73-81)) - ((N73-877) - (R75-12))      -206.444  94.9 94
##  (Centennial - (R73-81)) - ((N73-877) - Tracy)          -12.444  94.9 94
##  (Centennial - (R73-81)) - ((N73-882) - (R73-81))        -1.889  67.1 94
##  (Centennial - (R73-81)) - ((N73-882) - (R75-12))      -199.778  94.9 94
##  (Centennial - (R73-81)) - ((N73-882) - Tracy)           -5.778  94.9 94
##  (Centennial - (R73-81)) - ((R73-81) - (R75-12))       -176.889 116.2 94
##  (Centennial - (R73-81)) - ((R73-81) - Tracy)            17.111 116.2 94
##  (Centennial - (R73-81)) - ((R75-12) - Tracy)           215.000  94.9 94
##  (Centennial - (R75-12)) - (Centennial - Tracy)         194.000  67.1 94
##  (Centennial - (R75-12)) - ((D74-7741) - (N72-137))     296.333  94.9 94
##  (Centennial - (R75-12)) - ((D74-7741) - (N72-3058))    143.111  94.9 94
##  (Centennial - (R75-12)) - ((D74-7741) - (N72-3148))    378.444  94.9 94
##  (Centennial - (R75-12)) - ((D74-7741) - (N73-1102))    313.889  94.9 94
##  (Centennial - (R75-12)) - ((D74-7741) - (N73-693))     248.667  94.9 94
##  (Centennial - (R75-12)) - ((D74-7741) - (N73-877))     215.778  94.9 94
##  (Centennial - (R75-12)) - ((D74-7741) - (N73-882))     209.111  94.9 94
##  (Centennial - (R75-12)) - ((D74-7741) - (R73-81))      186.222  94.9 94
##  (Centennial - (R75-12)) - ((D74-7741) - (R75-12))      -11.667  67.1 94
##  (Centennial - (R75-12)) - ((D74-7741) - Tracy)         182.333  94.9 94
##  (Centennial - (R75-12)) - ((N72-137) - (N72-3058))      65.667  94.9 94
##  (Centennial - (R75-12)) - ((N72-137) - (N72-3148))     301.000  94.9 94
##  (Centennial - (R75-12)) - ((N72-137) - (N73-1102))     236.444  94.9 94
##  (Centennial - (R75-12)) - ((N72-137) - (N73-693))      171.222  94.9 94
##  (Centennial - (R75-12)) - ((N72-137) - (N73-877))      138.333  94.9 94
##  (Centennial - (R75-12)) - ((N72-137) - (N73-882))      131.667  94.9 94
##  (Centennial - (R75-12)) - ((N72-137) - (R73-81))       108.778  94.9 94
##  (Centennial - (R75-12)) - ((N72-137) - (R75-12))       -89.111  67.1 94
##  (Centennial - (R75-12)) - ((N72-137) - Tracy)          104.889  94.9 94
##  (Centennial - (R75-12)) - ((N72-3058) - (N72-3148))    454.222  94.9 94
##  (Centennial - (R75-12)) - ((N72-3058) - (N73-1102))    389.667  94.9 94
##  (Centennial - (R75-12)) - ((N72-3058) - (N73-693))     324.444  94.9 94
##  (Centennial - (R75-12)) - ((N72-3058) - (N73-877))     291.556  94.9 94
##  (Centennial - (R75-12)) - ((N72-3058) - (N73-882))     284.889  94.9 94
##  (Centennial - (R75-12)) - ((N72-3058) - (R73-81))      262.000  94.9 94
##  (Centennial - (R75-12)) - ((N72-3058) - (R75-12))       64.111  67.1 94
##  (Centennial - (R75-12)) - ((N72-3058) - Tracy)         258.111  94.9 94
##  (Centennial - (R75-12)) - ((N72-3148) - (N73-1102))    154.333  94.9 94
##  (Centennial - (R75-12)) - ((N72-3148) - (N73-693))      89.111  94.9 94
##  (Centennial - (R75-12)) - ((N72-3148) - (N73-877))      56.222  94.9 94
##  (Centennial - (R75-12)) - ((N72-3148) - (N73-882))      49.556  94.9 94
##  (Centennial - (R75-12)) - ((N72-3148) - (R73-81))       26.667  94.9 94
##  (Centennial - (R75-12)) - ((N72-3148) - (R75-12))     -171.222  67.1 94
##  (Centennial - (R75-12)) - ((N72-3148) - Tracy)          22.778  94.9 94
##  (Centennial - (R75-12)) - ((N73-1102) - (N73-693))     153.667  94.9 94
##  (Centennial - (R75-12)) - ((N73-1102) - (N73-877))     120.778  94.9 94
##  (Centennial - (R75-12)) - ((N73-1102) - (N73-882))     114.111  94.9 94
##  (Centennial - (R75-12)) - ((N73-1102) - (R73-81))       91.222  94.9 94
##  (Centennial - (R75-12)) - ((N73-1102) - (R75-12))     -106.667  67.1 94
##  (Centennial - (R75-12)) - ((N73-1102) - Tracy)          87.333  94.9 94
##  (Centennial - (R75-12)) - ((N73-693) - (N73-877))      186.000  94.9 94
##  (Centennial - (R75-12)) - ((N73-693) - (N73-882))      179.333  94.9 94
##  (Centennial - (R75-12)) - ((N73-693) - (R73-81))       156.444  94.9 94
##  (Centennial - (R75-12)) - ((N73-693) - (R75-12))       -41.444  67.1 94
##  (Centennial - (R75-12)) - ((N73-693) - Tracy)          152.556  94.9 94
##  (Centennial - (R75-12)) - ((N73-877) - (N73-882))      212.222  94.9 94
##  (Centennial - (R75-12)) - ((N73-877) - (R73-81))       189.333  94.9 94
##  (Centennial - (R75-12)) - ((N73-877) - (R75-12))        -8.556  67.1 94
##  (Centennial - (R75-12)) - ((N73-877) - Tracy)          185.444  94.9 94
##  (Centennial - (R75-12)) - ((N73-882) - (R73-81))       196.000  94.9 94
##  (Centennial - (R75-12)) - ((N73-882) - (R75-12))        -1.889  67.1 94
##  (Centennial - (R75-12)) - ((N73-882) - Tracy)          192.111  94.9 94
##  (Centennial - (R75-12)) - ((R73-81) - (R75-12))         21.000  67.1 94
##  (Centennial - (R75-12)) - ((R73-81) - Tracy)           215.000  94.9 94
##  (Centennial - (R75-12)) - ((R75-12) - Tracy)           412.889 116.2 94
##  (Centennial - Tracy) - ((D74-7741) - (N72-137))        102.333  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - (N72-3058))       -50.889  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - (N72-3148))       184.444  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - (N73-1102))       119.889  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - (N73-693))         54.667  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - (N73-877))         21.778  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - (N73-882))         15.111  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - (R73-81))          -7.778  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - (R75-12))        -205.667  94.9 94
##  (Centennial - Tracy) - ((D74-7741) - Tracy)            -11.667  67.1 94
##  (Centennial - Tracy) - ((N72-137) - (N72-3058))       -128.333  94.9 94
##  (Centennial - Tracy) - ((N72-137) - (N72-3148))        107.000  94.9 94
##  (Centennial - Tracy) - ((N72-137) - (N73-1102))         42.444  94.9 94
##  (Centennial - Tracy) - ((N72-137) - (N73-693))         -22.778  94.9 94
##  (Centennial - Tracy) - ((N72-137) - (N73-877))         -55.667  94.9 94
##  (Centennial - Tracy) - ((N72-137) - (N73-882))         -62.333  94.9 94
##  (Centennial - Tracy) - ((N72-137) - (R73-81))          -85.222  94.9 94
##  (Centennial - Tracy) - ((N72-137) - (R75-12))         -283.111  94.9 94
##  (Centennial - Tracy) - ((N72-137) - Tracy)             -89.111  67.1 94
##  (Centennial - Tracy) - ((N72-3058) - (N72-3148))       260.222  94.9 94
##  (Centennial - Tracy) - ((N72-3058) - (N73-1102))       195.667  94.9 94
##  (Centennial - Tracy) - ((N72-3058) - (N73-693))        130.444  94.9 94
##  (Centennial - Tracy) - ((N72-3058) - (N73-877))         97.556  94.9 94
##  (Centennial - Tracy) - ((N72-3058) - (N73-882))         90.889  94.9 94
##  (Centennial - Tracy) - ((N72-3058) - (R73-81))          68.000  94.9 94
##  (Centennial - Tracy) - ((N72-3058) - (R75-12))        -129.889  94.9 94
##  (Centennial - Tracy) - ((N72-3058) - Tracy)             64.111  67.1 94
##  (Centennial - Tracy) - ((N72-3148) - (N73-1102))       -39.667  94.9 94
##  (Centennial - Tracy) - ((N72-3148) - (N73-693))       -104.889  94.9 94
##  (Centennial - Tracy) - ((N72-3148) - (N73-877))       -137.778  94.9 94
##  (Centennial - Tracy) - ((N72-3148) - (N73-882))       -144.444  94.9 94
##  (Centennial - Tracy) - ((N72-3148) - (R73-81))        -167.333  94.9 94
##  (Centennial - Tracy) - ((N72-3148) - (R75-12))        -365.222  94.9 94
##  (Centennial - Tracy) - ((N72-3148) - Tracy)           -171.222  67.1 94
##  (Centennial - Tracy) - ((N73-1102) - (N73-693))        -40.333  94.9 94
##  (Centennial - Tracy) - ((N73-1102) - (N73-877))        -73.222  94.9 94
##  (Centennial - Tracy) - ((N73-1102) - (N73-882))        -79.889  94.9 94
##  (Centennial - Tracy) - ((N73-1102) - (R73-81))        -102.778  94.9 94
##  (Centennial - Tracy) - ((N73-1102) - (R75-12))        -300.667  94.9 94
##  (Centennial - Tracy) - ((N73-1102) - Tracy)           -106.667  67.1 94
##  (Centennial - Tracy) - ((N73-693) - (N73-877))          -8.000  94.9 94
##  (Centennial - Tracy) - ((N73-693) - (N73-882))         -14.667  94.9 94
##  (Centennial - Tracy) - ((N73-693) - (R73-81))          -37.556  94.9 94
##  (Centennial - Tracy) - ((N73-693) - (R75-12))         -235.444  94.9 94
##  (Centennial - Tracy) - ((N73-693) - Tracy)             -41.444  67.1 94
##  (Centennial - Tracy) - ((N73-877) - (N73-882))          18.222  94.9 94
##  (Centennial - Tracy) - ((N73-877) - (R73-81))           -4.667  94.9 94
##  (Centennial - Tracy) - ((N73-877) - (R75-12))         -202.556  94.9 94
##  (Centennial - Tracy) - ((N73-877) - Tracy)              -8.556  67.1 94
##  (Centennial - Tracy) - ((N73-882) - (R73-81))            2.000  94.9 94
##  (Centennial - Tracy) - ((N73-882) - (R75-12))         -195.889  94.9 94
##  (Centennial - Tracy) - ((N73-882) - Tracy)              -1.889  67.1 94
##  (Centennial - Tracy) - ((R73-81) - (R75-12))          -173.000  94.9 94
##  (Centennial - Tracy) - ((R73-81) - Tracy)               21.000  67.1 94
##  (Centennial - Tracy) - ((R75-12) - Tracy)              218.889  67.1 94
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##  ((D74-7741) - (N72-137)) - ((D74-7741) - (N73-1102))    17.556  67.1 94
##  ((D74-7741) - (N72-137)) - ((D74-7741) - (N73-693))    -47.667  67.1 94
##  ((D74-7741) - (N72-137)) - ((D74-7741) - (N73-877))    -80.556  67.1 94
##  ((D74-7741) - (N72-137)) - ((D74-7741) - (N73-882))    -87.222  67.1 94
##  ((D74-7741) - (N72-137)) - ((D74-7741) - (R73-81))    -110.111  67.1 94
##  ((D74-7741) - (N72-137)) - ((D74-7741) - (R75-12))    -308.000  67.1 94
##  ((D74-7741) - (N72-137)) - ((D74-7741) - Tracy)       -114.000  67.1 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - (N72-3058))   -230.667 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - (N72-3148))      4.667 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - (N73-1102))    -59.889 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - (N73-693))    -125.111 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - (N73-877))    -158.000 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - (N73-882))    -164.667 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - (R73-81))     -187.556 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - (R75-12))     -385.444 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-137) - Tracy)        -191.444 116.2 94
##  ((D74-7741) - (N72-137)) - ((N72-3058) - (N72-3148))   157.889  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3058) - (N73-1102))    93.333  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3058) - (N73-693))     28.111  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3058) - (N73-877))     -4.778  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3058) - (N73-882))    -11.444  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3058) - (R73-81))     -34.333  94.9 94
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##  ((D74-7741) - (N72-137)) - ((N72-3148) - (N73-693))   -207.222  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3148) - (N73-877))   -240.111  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3148) - (N73-882))   -246.778  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3148) - (R73-81))    -269.667  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3148) - (R75-12))    -467.556  94.9 94
##  ((D74-7741) - (N72-137)) - ((N72-3148) - Tracy)       -273.556  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-1102) - (N73-693))   -142.667  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-1102) - (N73-877))   -175.556  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-1102) - (N73-882))   -182.222  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-1102) - (R73-81))    -205.111  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-1102) - (R75-12))    -403.000  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-1102) - Tracy)       -209.000  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-693) - (N73-877))    -110.333  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-693) - (N73-882))    -117.000  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-693) - (R73-81))     -139.889  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-693) - (R75-12))     -337.778  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-693) - Tracy)        -143.778  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-877) - (N73-882))     -84.111  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-877) - (R73-81))     -107.000  94.9 94
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##  ((D74-7741) - (N72-137)) - ((N73-877) - Tracy)        -110.889  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-882) - (R73-81))     -100.333  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-882) - (R75-12))     -298.222  94.9 94
##  ((D74-7741) - (N72-137)) - ((N73-882) - Tracy)        -104.222  94.9 94
##  ((D74-7741) - (N72-137)) - ((R73-81) - (R75-12))      -275.333  94.9 94
##  ((D74-7741) - (N72-137)) - ((R73-81) - Tracy)          -81.333  94.9 94
##  ((D74-7741) - (N72-137)) - ((R75-12) - Tracy)          116.556  94.9 94
##  ((D74-7741) - (N72-3058)) - ((D74-7741) - (N72-3148))  235.333  67.1 94
##  ((D74-7741) - (N72-3058)) - ((D74-7741) - (N73-1102))  170.778  67.1 94
##  ((D74-7741) - (N72-3058)) - ((D74-7741) - (N73-693))   105.556  67.1 94
##  ((D74-7741) - (N72-3058)) - ((D74-7741) - (N73-877))    72.667  67.1 94
##  ((D74-7741) - (N72-3058)) - ((D74-7741) - (N73-882))    66.000  67.1 94
##  ((D74-7741) - (N72-3058)) - ((D74-7741) - (R73-81))     43.111  67.1 94
##  ((D74-7741) - (N72-3058)) - ((D74-7741) - (R75-12))   -154.778  67.1 94
##  ((D74-7741) - (N72-3058)) - ((D74-7741) - Tracy)        39.222  67.1 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - (N72-3058))   -77.444  67.1 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - (N72-3148))   157.889  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - (N73-1102))    93.333  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - (N73-693))     28.111  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - (N73-877))     -4.778  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - (N73-882))    -11.444  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - (R73-81))     -34.333  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - (R75-12))    -232.222  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-137) - Tracy)        -38.222  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-3058) - (N72-3148))  311.111 116.2 94
##  ((D74-7741) - (N72-3058)) - ((N72-3058) - (N73-1102))  246.556 116.2 94
##  ((D74-7741) - (N72-3058)) - ((N72-3058) - (N73-693))   181.333 116.2 94
##  ((D74-7741) - (N72-3058)) - ((N72-3058) - (N73-877))   148.444 116.2 94
##  ((D74-7741) - (N72-3058)) - ((N72-3058) - (N73-882))   141.778 116.2 94
##  ((D74-7741) - (N72-3058)) - ((N72-3058) - (R73-81))    118.889 116.2 94
##  ((D74-7741) - (N72-3058)) - ((N72-3058) - (R75-12))    -79.000 116.2 94
##  ((D74-7741) - (N72-3058)) - ((N72-3058) - Tracy)       115.000 116.2 94
##  ((D74-7741) - (N72-3058)) - ((N72-3148) - (N73-1102))   11.222  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-3148) - (N73-693))   -54.000  94.9 94
##  ((D74-7741) - (N72-3058)) - ((N72-3148) - (N73-877))   -86.889  94.9 94
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##  ((D74-7741) - (N73-882)) - ((N72-3058) - (N73-693))    115.333  94.9 94
##  ((D74-7741) - (N73-882)) - ((N72-3058) - (N73-877))     82.444  94.9 94
##  ((D74-7741) - (N73-882)) - ((N72-3058) - (N73-882))     75.778  67.1 94
##  ((D74-7741) - (N73-882)) - ((N72-3058) - (R73-81))      52.889  94.9 94
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##  ((D74-7741) - (N73-882)) - ((R73-81) - Tracy)            5.889  94.9 94
##  ((D74-7741) - (N73-882)) - ((R75-12) - Tracy)          203.778  94.9 94
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##  ((D74-7741) - (R73-81)) - ((N72-3058) - (N72-3148))    268.000  94.9 94
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##  ((D74-7741) - (R73-81)) - ((N72-3058) - (N73-693))     138.222  94.9 94
##  ((D74-7741) - (R73-81)) - ((N72-3058) - (N73-877))     105.333  94.9 94
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##  ((D74-7741) - (R73-81)) - ((N72-3058) - (R73-81))       75.778  67.1 94
##  ((D74-7741) - (R73-81)) - ((N72-3058) - (R75-12))     -122.111  94.9 94
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##  ((D74-7741) - (R73-81)) - ((N73-693) - (N73-877))       -0.222  94.9 94
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##  ((D74-7741) - (R73-81)) - ((N73-693) - (R73-81))       -29.778  67.1 94
##  ((D74-7741) - (R73-81)) - ((N73-693) - (R75-12))      -227.667  94.9 94
##  ((D74-7741) - (R73-81)) - ((N73-693) - Tracy)          -33.667  94.9 94
##  ((D74-7741) - (R73-81)) - ((N73-877) - (N73-882))       26.000  94.9 94
##  ((D74-7741) - (R73-81)) - ((N73-877) - (R73-81))         3.111  67.1 94
##  ((D74-7741) - (R73-81)) - ((N73-877) - (R75-12))      -194.778  94.9 94
##  ((D74-7741) - (R73-81)) - ((N73-877) - Tracy)           -0.778  94.9 94
##  ((D74-7741) - (R73-81)) - ((N73-882) - (R73-81))         9.778  67.1 94
##  ((D74-7741) - (R73-81)) - ((N73-882) - (R75-12))      -188.111  94.9 94
##  ((D74-7741) - (R73-81)) - ((N73-882) - Tracy)            5.889  94.9 94
##  ((D74-7741) - (R73-81)) - ((R73-81) - (R75-12))       -165.222 116.2 94
##  ((D74-7741) - (R73-81)) - ((R73-81) - Tracy)            28.778 116.2 94
##  ((D74-7741) - (R73-81)) - ((R75-12) - Tracy)           226.667  94.9 94
##  ((D74-7741) - (R75-12)) - ((D74-7741) - Tracy)         194.000  67.1 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - (N72-3058))      77.333  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - (N72-3148))     312.667  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - (N73-1102))     248.111  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - (N73-693))      182.889  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - (N73-877))      150.000  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - (N73-882))      143.333  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - (R73-81))       120.444  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - (R75-12))       -77.444  67.1 94
##  ((D74-7741) - (R75-12)) - ((N72-137) - Tracy)          116.556  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3058) - (N72-3148))    465.889  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3058) - (N73-1102))    401.333  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3058) - (N73-693))     336.111  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3058) - (N73-877))     303.222  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3058) - (N73-882))     296.556  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3058) - (R73-81))      273.667  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3058) - (R75-12))       75.778  67.1 94
##  ((D74-7741) - (R75-12)) - ((N72-3058) - Tracy)         269.778  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3148) - (N73-1102))    166.000  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3148) - (N73-693))     100.778  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3148) - (N73-877))      67.889  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3148) - (N73-882))      61.222  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3148) - (R73-81))       38.333  94.9 94
##  ((D74-7741) - (R75-12)) - ((N72-3148) - (R75-12))     -159.556  67.1 94
##  ((D74-7741) - (R75-12)) - ((N72-3148) - Tracy)          34.444  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-1102) - (N73-693))     165.333  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-1102) - (N73-877))     132.444  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-1102) - (N73-882))     125.778  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-1102) - (R73-81))      102.889  94.9 94
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##  ((D74-7741) - (R75-12)) - ((N73-1102) - Tracy)          99.000  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-693) - (N73-877))      197.667  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-693) - (N73-882))      191.000  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-693) - (R73-81))       168.111  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-693) - (R75-12))       -29.778  67.1 94
##  ((D74-7741) - (R75-12)) - ((N73-693) - Tracy)          164.222  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-877) - (N73-882))      223.889  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-877) - (R73-81))       201.000  94.9 94
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##  ((D74-7741) - (R75-12)) - ((N73-877) - Tracy)          197.111  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-882) - (R73-81))       207.667  94.9 94
##  ((D74-7741) - (R75-12)) - ((N73-882) - (R75-12))         9.778  67.1 94
##  ((D74-7741) - (R75-12)) - ((N73-882) - Tracy)          203.778  94.9 94
##  ((D74-7741) - (R75-12)) - ((R73-81) - (R75-12))         32.667  67.1 94
##  ((D74-7741) - (R75-12)) - ((R73-81) - Tracy)           226.667  94.9 94
##  ((D74-7741) - (R75-12)) - ((R75-12) - Tracy)           424.556 116.2 94
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##  ((D74-7741) - Tracy) - ((N72-137) - (N72-3148))        118.667  94.9 94
##  ((D74-7741) - Tracy) - ((N72-137) - (N73-1102))         54.111  94.9 94
##  ((D74-7741) - Tracy) - ((N72-137) - (N73-693))         -11.111  94.9 94
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##  ((D74-7741) - Tracy) - ((N72-137) - Tracy)             -77.444  67.1 94
##  ((D74-7741) - Tracy) - ((N72-3058) - (N72-3148))       271.889  94.9 94
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##  ((D74-7741) - Tracy) - ((N72-3058) - (N73-693))        142.111  94.9 94
##  ((D74-7741) - Tracy) - ((N72-3058) - (N73-877))        109.222  94.9 94
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##  ((D74-7741) - Tracy) - ((N72-3058) - Tracy)             75.778  67.1 94
##  ((D74-7741) - Tracy) - ((N72-3148) - (N73-1102))       -28.000  94.9 94
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##  ((D74-7741) - Tracy) - ((N73-1102) - (N73-693))        -28.667  94.9 94
##  ((D74-7741) - Tracy) - ((N73-1102) - (N73-877))        -61.556  94.9 94
##  ((D74-7741) - Tracy) - ((N73-1102) - (N73-882))        -68.222  94.9 94
##  ((D74-7741) - Tracy) - ((N73-1102) - (R73-81))         -91.111  94.9 94
##  ((D74-7741) - Tracy) - ((N73-1102) - (R75-12))        -289.000  94.9 94
##  ((D74-7741) - Tracy) - ((N73-1102) - Tracy)            -95.000  67.1 94
##  ((D74-7741) - Tracy) - ((N73-693) - (N73-877))           3.667  94.9 94
##  ((D74-7741) - Tracy) - ((N73-693) - (N73-882))          -3.000  94.9 94
##  ((D74-7741) - Tracy) - ((N73-693) - (R73-81))          -25.889  94.9 94
##  ((D74-7741) - Tracy) - ((N73-693) - (R75-12))         -223.778  94.9 94
##  ((D74-7741) - Tracy) - ((N73-693) - Tracy)             -29.778  67.1 94
##  ((D74-7741) - Tracy) - ((N73-877) - (N73-882))          29.889  94.9 94
##  ((D74-7741) - Tracy) - ((N73-877) - (R73-81))            7.000  94.9 94
##  ((D74-7741) - Tracy) - ((N73-877) - (R75-12))         -190.889  94.9 94
##  ((D74-7741) - Tracy) - ((N73-877) - Tracy)               3.111  67.1 94
##  ((D74-7741) - Tracy) - ((N73-882) - (R73-81))           13.667  94.9 94
##  ((D74-7741) - Tracy) - ((N73-882) - (R75-12))         -184.222  94.9 94
##  ((D74-7741) - Tracy) - ((N73-882) - Tracy)               9.778  67.1 94
##  ((D74-7741) - Tracy) - ((R73-81) - (R75-12))          -161.333  94.9 94
##  ((D74-7741) - Tracy) - ((R73-81) - Tracy)               32.667  67.1 94
##  ((D74-7741) - Tracy) - ((R75-12) - Tracy)              230.556  67.1 94
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##  ((N72-137) - (N72-3058)) - ((N72-137) - (N73-693))     105.556  67.1 94
##  ((N72-137) - (N72-3058)) - ((N72-137) - (N73-877))      72.667  67.1 94
##  ((N72-137) - (N72-3058)) - ((N72-137) - (N73-882))      66.000  67.1 94
##  ((N72-137) - (N72-3058)) - ((N72-137) - (R73-81))       43.111  67.1 94
##  ((N72-137) - (N72-3058)) - ((N72-137) - (R75-12))     -154.778  67.1 94
##  ((N72-137) - (N72-3058)) - ((N72-137) - Tracy)          39.222  67.1 94
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##  ((N72-137) - (N72-3058)) - ((N72-3058) - (N73-1102))   324.000 116.2 94
##  ((N72-137) - (N72-3058)) - ((N72-3058) - (N73-693))    258.778 116.2 94
##  ((N72-137) - (N72-3058)) - ((N72-3058) - (N73-877))    225.889 116.2 94
##  ((N72-137) - (N72-3058)) - ((N72-3058) - (N73-882))    219.222 116.2 94
##  ((N72-137) - (N72-3058)) - ((N72-3058) - (R73-81))     196.333 116.2 94
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##  ((N72-137) - (N72-3058)) - ((N72-3058) - Tracy)        192.444 116.2 94
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##  ((N72-137) - (N72-3058)) - ((N73-693) - (R73-81))       90.778  94.9 94
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##  ((N72-137) - (N72-3058)) - ((N73-877) - (R73-81))      123.667  94.9 94
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##  ((N72-137) - (N72-3058)) - ((N73-882) - (R75-12))      -67.556  94.9 94
##  ((N72-137) - (N72-3058)) - ((N73-882) - Tracy)         126.444  94.9 94
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##  ((N72-137) - (N72-3058)) - ((R73-81) - Tracy)          149.333  94.9 94
##  ((N72-137) - (N72-3058)) - ((R75-12) - Tracy)          347.222  94.9 94
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##  ((N72-137) - (N72-3148)) - ((N72-3058) - (N73-693))     23.444  94.9 94
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##  ((N72-137) - (N73-1102)) - ((N72-3058) - (N73-693))     88.000  94.9 94
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##  ((N72-137) - (N73-1102)) - ((N72-3058) - (N73-882))     48.444  94.9 94
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##  ((N72-137) - (N73-693)) - ((N72-3058) - (N72-3148))    283.000  94.9 94
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##  ((N72-137) - (N73-877)) - ((N72-3058) - (R75-12))      -74.222  94.9 94
##  ((N72-137) - (N73-877)) - ((N72-3058) - Tracy)         119.778  94.9 94
##  ((N72-137) - (N73-877)) - ((N72-3148) - (N73-1102))     16.000  94.9 94
##  ((N72-137) - (N73-877)) - ((N72-3148) - (N73-693))     -49.222  94.9 94
##  ((N72-137) - (N73-877)) - ((N72-3148) - (N73-877))     -82.111  67.1 94
##  ((N72-137) - (N73-877)) - ((N72-3148) - (N73-882))     -88.778  94.9 94
##  ((N72-137) - (N73-877)) - ((N72-3148) - (R73-81))     -111.667  94.9 94
##  ((N72-137) - (N73-877)) - ((N72-3148) - (R75-12))     -309.556  94.9 94
##  ((N72-137) - (N73-877)) - ((N72-3148) - Tracy)        -115.556  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-1102) - (N73-693))      15.333  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-1102) - (N73-877))     -17.556  67.1 94
##  ((N72-137) - (N73-877)) - ((N73-1102) - (N73-882))     -24.222  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-1102) - (R73-81))      -47.111  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-1102) - (R75-12))     -245.000  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-1102) - Tracy)         -51.000  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-693) - (N73-877))       47.667  67.1 94
##  ((N72-137) - (N73-877)) - ((N73-693) - (N73-882))       41.000  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-693) - (R73-81))        18.111  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-693) - (R75-12))      -179.778  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-693) - Tracy)           14.222  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-877) - (N73-882))       73.889 116.2 94
##  ((N72-137) - (N73-877)) - ((N73-877) - (R73-81))        51.000 116.2 94
##  ((N72-137) - (N73-877)) - ((N73-877) - (R75-12))      -146.889 116.2 94
##  ((N72-137) - (N73-877)) - ((N73-877) - Tracy)           47.111 116.2 94
##  ((N72-137) - (N73-877)) - ((N73-882) - (R73-81))        57.667  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-882) - (R75-12))      -140.222  94.9 94
##  ((N72-137) - (N73-877)) - ((N73-882) - Tracy)           53.778  94.9 94
##  ((N72-137) - (N73-877)) - ((R73-81) - (R75-12))       -117.333  94.9 94
##  ((N72-137) - (N73-877)) - ((R73-81) - Tracy)            76.667  94.9 94
##  ((N72-137) - (N73-877)) - ((R75-12) - Tracy)           274.556  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-137) - (R73-81))       -22.889  67.1 94
##  ((N72-137) - (N73-882)) - ((N72-137) - (R75-12))      -220.778  67.1 94
##  ((N72-137) - (N73-882)) - ((N72-137) - Tracy)          -26.778  67.1 94
##  ((N72-137) - (N73-882)) - ((N72-3058) - (N72-3148))    322.556  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3058) - (N73-1102))    258.000  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3058) - (N73-693))     192.778  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3058) - (N73-877))     159.889  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3058) - (N73-882))     153.222  67.1 94
##  ((N72-137) - (N73-882)) - ((N72-3058) - (R73-81))      130.333  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3058) - (R75-12))      -67.556  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3058) - Tracy)         126.444  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3148) - (N73-1102))     22.667  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3148) - (N73-693))     -42.556  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3148) - (N73-877))     -75.444  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3148) - (N73-882))     -82.111  67.1 94
##  ((N72-137) - (N73-882)) - ((N72-3148) - (R73-81))     -105.000  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3148) - (R75-12))     -302.889  94.9 94
##  ((N72-137) - (N73-882)) - ((N72-3148) - Tracy)        -108.889  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-1102) - (N73-693))      22.000  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-1102) - (N73-877))     -10.889  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-1102) - (N73-882))     -17.556  67.1 94
##  ((N72-137) - (N73-882)) - ((N73-1102) - (R73-81))      -40.444  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-1102) - (R75-12))     -238.333  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-1102) - Tracy)         -44.333  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-693) - (N73-877))       54.333  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-693) - (N73-882))       47.667  67.1 94
##  ((N72-137) - (N73-882)) - ((N73-693) - (R73-81))        24.778  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-693) - (R75-12))      -173.111  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-693) - Tracy)           20.889  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-877) - (N73-882))       80.556  67.1 94
##  ((N72-137) - (N73-882)) - ((N73-877) - (R73-81))        57.667  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-877) - (R75-12))      -140.222  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-877) - Tracy)           53.778  94.9 94
##  ((N72-137) - (N73-882)) - ((N73-882) - (R73-81))        64.333 116.2 94
##  ((N72-137) - (N73-882)) - ((N73-882) - (R75-12))      -133.556 116.2 94
##  ((N72-137) - (N73-882)) - ((N73-882) - Tracy)           60.444 116.2 94
##  ((N72-137) - (N73-882)) - ((R73-81) - (R75-12))       -110.667  94.9 94
##  ((N72-137) - (N73-882)) - ((R73-81) - Tracy)            83.333  94.9 94
##  ((N72-137) - (N73-882)) - ((R75-12) - Tracy)           281.222  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-137) - (R75-12))       -197.889  67.1 94
##  ((N72-137) - (R73-81)) - ((N72-137) - Tracy)            -3.889  67.1 94
##  ((N72-137) - (R73-81)) - ((N72-3058) - (N72-3148))     345.444  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3058) - (N73-1102))     280.889  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3058) - (N73-693))      215.667  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3058) - (N73-877))      182.778  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3058) - (N73-882))      176.111  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3058) - (R73-81))       153.222  67.1 94
##  ((N72-137) - (R73-81)) - ((N72-3058) - (R75-12))       -44.667  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3058) - Tracy)          149.333  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3148) - (N73-1102))      45.556  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3148) - (N73-693))      -19.667  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3148) - (N73-877))      -52.556  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3148) - (N73-882))      -59.222  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3148) - (R73-81))       -82.111  67.1 94
##  ((N72-137) - (R73-81)) - ((N72-3148) - (R75-12))      -280.000  94.9 94
##  ((N72-137) - (R73-81)) - ((N72-3148) - Tracy)          -86.000  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-1102) - (N73-693))       44.889  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-1102) - (N73-877))       12.000  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-1102) - (N73-882))        5.333  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-1102) - (R73-81))       -17.556  67.1 94
##  ((N72-137) - (R73-81)) - ((N73-1102) - (R75-12))      -215.444  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-1102) - Tracy)          -21.444  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-693) - (N73-877))        77.222  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-693) - (N73-882))        70.556  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-693) - (R73-81))         47.667  67.1 94
##  ((N72-137) - (R73-81)) - ((N73-693) - (R75-12))       -150.222  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-693) - Tracy)            43.778  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-877) - (N73-882))       103.444  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-877) - (R73-81))         80.556  67.1 94
##  ((N72-137) - (R73-81)) - ((N73-877) - (R75-12))       -117.333  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-877) - Tracy)            76.667  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-882) - (R73-81))         87.222  67.1 94
##  ((N72-137) - (R73-81)) - ((N73-882) - (R75-12))       -110.667  94.9 94
##  ((N72-137) - (R73-81)) - ((N73-882) - Tracy)            83.333  94.9 94
##  ((N72-137) - (R73-81)) - ((R73-81) - (R75-12))         -87.778 116.2 94
##  ((N72-137) - (R73-81)) - ((R73-81) - Tracy)            106.222 116.2 94
##  ((N72-137) - (R73-81)) - ((R75-12) - Tracy)            304.111  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-137) - Tracy)           194.000  67.1 94
##  ((N72-137) - (R75-12)) - ((N72-3058) - (N72-3148))     543.333  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3058) - (N73-1102))     478.778  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3058) - (N73-693))      413.556  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3058) - (N73-877))      380.667  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3058) - (N73-882))      374.000  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3058) - (R73-81))       351.111  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3058) - (R75-12))       153.222  67.1 94
##  ((N72-137) - (R75-12)) - ((N72-3058) - Tracy)          347.222  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3148) - (N73-1102))     243.444  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3148) - (N73-693))      178.222  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3148) - (N73-877))      145.333  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3148) - (N73-882))      138.667  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3148) - (R73-81))       115.778  94.9 94
##  ((N72-137) - (R75-12)) - ((N72-3148) - (R75-12))       -82.111  67.1 94
##  ((N72-137) - (R75-12)) - ((N72-3148) - Tracy)          111.889  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-1102) - (N73-693))      242.778  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-1102) - (N73-877))      209.889  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-1102) - (N73-882))      203.222  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-1102) - (R73-81))       180.333  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-1102) - (R75-12))       -17.556  67.1 94
##  ((N72-137) - (R75-12)) - ((N73-1102) - Tracy)          176.444  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-693) - (N73-877))       275.111  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-693) - (N73-882))       268.444  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-693) - (R73-81))        245.556  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-693) - (R75-12))         47.667  67.1 94
##  ((N72-137) - (R75-12)) - ((N73-693) - Tracy)           241.667  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-877) - (N73-882))       301.333  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-877) - (R73-81))        278.444  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-877) - (R75-12))         80.556  67.1 94
##  ((N72-137) - (R75-12)) - ((N73-877) - Tracy)           274.556  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-882) - (R73-81))        285.111  94.9 94
##  ((N72-137) - (R75-12)) - ((N73-882) - (R75-12))         87.222  67.1 94
##  ((N72-137) - (R75-12)) - ((N73-882) - Tracy)           281.222  94.9 94
##  ((N72-137) - (R75-12)) - ((R73-81) - (R75-12))         110.111  67.1 94
##  ((N72-137) - (R75-12)) - ((R73-81) - Tracy)            304.111  94.9 94
##  ((N72-137) - (R75-12)) - ((R75-12) - Tracy)            502.000 116.2 94
##  ((N72-137) - Tracy) - ((N72-3058) - (N72-3148))        349.333  94.9 94
##  ((N72-137) - Tracy) - ((N72-3058) - (N73-1102))        284.778  94.9 94
##  ((N72-137) - Tracy) - ((N72-3058) - (N73-693))         219.556  94.9 94
##  ((N72-137) - Tracy) - ((N72-3058) - (N73-877))         186.667  94.9 94
##  ((N72-137) - Tracy) - ((N72-3058) - (N73-882))         180.000  94.9 94
##  ((N72-137) - Tracy) - ((N72-3058) - (R73-81))          157.111  94.9 94
##  ((N72-137) - Tracy) - ((N72-3058) - (R75-12))          -40.778  94.9 94
##  ((N72-137) - Tracy) - ((N72-3058) - Tracy)             153.222  67.1 94
##  ((N72-137) - Tracy) - ((N72-3148) - (N73-1102))         49.444  94.9 94
##  ((N72-137) - Tracy) - ((N72-3148) - (N73-693))         -15.778  94.9 94
##  ((N72-137) - Tracy) - ((N72-3148) - (N73-877))         -48.667  94.9 94
##  ((N72-137) - Tracy) - ((N72-3148) - (N73-882))         -55.333  94.9 94
##  ((N72-137) - Tracy) - ((N72-3148) - (R73-81))          -78.222  94.9 94
##  ((N72-137) - Tracy) - ((N72-3148) - (R75-12))         -276.111  94.9 94
##  ((N72-137) - Tracy) - ((N72-3148) - Tracy)             -82.111  67.1 94
##  ((N72-137) - Tracy) - ((N73-1102) - (N73-693))          48.778  94.9 94
##  ((N72-137) - Tracy) - ((N73-1102) - (N73-877))          15.889  94.9 94
##  ((N72-137) - Tracy) - ((N73-1102) - (N73-882))           9.222  94.9 94
##  ((N72-137) - Tracy) - ((N73-1102) - (R73-81))          -13.667  94.9 94
##  ((N72-137) - Tracy) - ((N73-1102) - (R75-12))         -211.556  94.9 94
##  ((N72-137) - Tracy) - ((N73-1102) - Tracy)             -17.556  67.1 94
##  ((N72-137) - Tracy) - ((N73-693) - (N73-877))           81.111  94.9 94
##  ((N72-137) - Tracy) - ((N73-693) - (N73-882))           74.444  94.9 94
##  ((N72-137) - Tracy) - ((N73-693) - (R73-81))            51.556  94.9 94
##  ((N72-137) - Tracy) - ((N73-693) - (R75-12))          -146.333  94.9 94
##  ((N72-137) - Tracy) - ((N73-693) - Tracy)               47.667  67.1 94
##  ((N72-137) - Tracy) - ((N73-877) - (N73-882))          107.333  94.9 94
##  ((N72-137) - Tracy) - ((N73-877) - (R73-81))            84.444  94.9 94
##  ((N72-137) - Tracy) - ((N73-877) - (R75-12))          -113.444  94.9 94
##  ((N72-137) - Tracy) - ((N73-877) - Tracy)               80.556  67.1 94
##  ((N72-137) - Tracy) - ((N73-882) - (R73-81))            91.111  94.9 94
##  ((N72-137) - Tracy) - ((N73-882) - (R75-12))          -106.778  94.9 94
##  ((N72-137) - Tracy) - ((N73-882) - Tracy)               87.222  67.1 94
##  ((N72-137) - Tracy) - ((R73-81) - (R75-12))            -83.889  94.9 94
##  ((N72-137) - Tracy) - ((R73-81) - Tracy)               110.111  67.1 94
##  ((N72-137) - Tracy) - ((R75-12) - Tracy)               308.000  67.1 94
##  ((N72-3058) - (N72-3148)) - ((N72-3058) - (N73-1102))  -64.556  67.1 94
##  ((N72-3058) - (N72-3148)) - ((N72-3058) - (N73-693))  -129.778  67.1 94
##  ((N72-3058) - (N72-3148)) - ((N72-3058) - (N73-877))  -162.667  67.1 94
##  ((N72-3058) - (N72-3148)) - ((N72-3058) - (N73-882))  -169.333  67.1 94
##  ((N72-3058) - (N72-3148)) - ((N72-3058) - (R73-81))   -192.222  67.1 94
##  ((N72-3058) - (N72-3148)) - ((N72-3058) - (R75-12))   -390.111  67.1 94
##  ((N72-3058) - (N72-3148)) - ((N72-3058) - Tracy)      -196.111  67.1 94
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##  ((N72-3058) - (N72-3148)) - ((N72-3148) - (N73-693))  -365.111 116.2 94
##  ((N72-3058) - (N72-3148)) - ((N72-3148) - (N73-877))  -398.000 116.2 94
##  ((N72-3058) - (N72-3148)) - ((N72-3148) - (N73-882))  -404.667 116.2 94
##  ((N72-3058) - (N72-3148)) - ((N72-3148) - (R73-81))   -427.556 116.2 94
##  ((N72-3058) - (N72-3148)) - ((N72-3148) - (R75-12))   -625.444 116.2 94
##  ((N72-3058) - (N72-3148)) - ((N72-3148) - Tracy)      -431.444 116.2 94
##  ((N72-3058) - (N72-3148)) - ((N73-1102) - (N73-693))  -300.556  94.9 94
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##  ((N72-3058) - (N72-3148)) - ((N73-1102) - (R75-12))   -560.889  94.9 94
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##  ((N72-3058) - (N72-3148)) - ((N73-693) - (N73-882))   -274.889  94.9 94
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##  ((N72-3058) - (R75-12)) - ((N73-882) - (R75-12))       -66.000  67.1 94
##  ((N72-3058) - (R75-12)) - ((N73-882) - Tracy)          128.000  94.9 94
##  ((N72-3058) - (R75-12)) - ((R73-81) - (R75-12))        -43.111  67.1 94
##  ((N72-3058) - (R75-12)) - ((R73-81) - Tracy)           150.889  94.9 94
##  ((N72-3058) - (R75-12)) - ((R75-12) - Tracy)           348.778 116.2 94
##  ((N72-3058) - Tracy) - ((N72-3148) - (N73-1102))      -103.778  94.9 94
##  ((N72-3058) - Tracy) - ((N72-3148) - (N73-693))       -169.000  94.9 94
##  ((N72-3058) - Tracy) - ((N72-3148) - (N73-877))       -201.889  94.9 94
##  ((N72-3058) - Tracy) - ((N72-3148) - (N73-882))       -208.556  94.9 94
##  ((N72-3058) - Tracy) - ((N72-3148) - (R73-81))        -231.444  94.9 94
##  ((N72-3058) - Tracy) - ((N72-3148) - (R75-12))        -429.333  94.9 94
##  ((N72-3058) - Tracy) - ((N72-3148) - Tracy)           -235.333  67.1 94
##  ((N72-3058) - Tracy) - ((N73-1102) - (N73-693))       -104.444  94.9 94
##  ((N72-3058) - Tracy) - ((N73-1102) - (N73-877))       -137.333  94.9 94
##  ((N72-3058) - Tracy) - ((N73-1102) - (N73-882))       -144.000  94.9 94
##  ((N72-3058) - Tracy) - ((N73-1102) - (R73-81))        -166.889  94.9 94
##  ((N72-3058) - Tracy) - ((N73-1102) - (R75-12))        -364.778  94.9 94
##  ((N72-3058) - Tracy) - ((N73-1102) - Tracy)           -170.778  67.1 94
##  ((N72-3058) - Tracy) - ((N73-693) - (N73-877))         -72.111  94.9 94
##  ((N72-3058) - Tracy) - ((N73-693) - (N73-882))         -78.778  94.9 94
##  ((N72-3058) - Tracy) - ((N73-693) - (R73-81))         -101.667  94.9 94
##  ((N72-3058) - Tracy) - ((N73-693) - (R75-12))         -299.556  94.9 94
##  ((N72-3058) - Tracy) - ((N73-693) - Tracy)            -105.556  67.1 94
##  ((N72-3058) - Tracy) - ((N73-877) - (N73-882))         -45.889  94.9 94
##  ((N72-3058) - Tracy) - ((N73-877) - (R73-81))          -68.778  94.9 94
##  ((N72-3058) - Tracy) - ((N73-877) - (R75-12))         -266.667  94.9 94
##  ((N72-3058) - Tracy) - ((N73-877) - Tracy)             -72.667  67.1 94
##  ((N72-3058) - Tracy) - ((N73-882) - (R73-81))          -62.111  94.9 94
##  ((N72-3058) - Tracy) - ((N73-882) - (R75-12))         -260.000  94.9 94
##  ((N72-3058) - Tracy) - ((N73-882) - Tracy)             -66.000  67.1 94
##  ((N72-3058) - Tracy) - ((R73-81) - (R75-12))          -237.111  94.9 94
##  ((N72-3058) - Tracy) - ((R73-81) - Tracy)              -43.111  67.1 94
##  ((N72-3058) - Tracy) - ((R75-12) - Tracy)              154.778  67.1 94
##  ((N72-3148) - (N73-1102)) - ((N72-3148) - (N73-693))   -65.222  67.1 94
##  ((N72-3148) - (N73-1102)) - ((N72-3148) - (N73-877))   -98.111  67.1 94
##  ((N72-3148) - (N73-1102)) - ((N72-3148) - (N73-882))  -104.778  67.1 94
##  ((N72-3148) - (N73-1102)) - ((N72-3148) - (R73-81))   -127.667  67.1 94
##  ((N72-3148) - (N73-1102)) - ((N72-3148) - (R75-12))   -325.556  67.1 94
##  ((N72-3148) - (N73-1102)) - ((N72-3148) - Tracy)      -131.556  67.1 94
##  ((N72-3148) - (N73-1102)) - ((N73-1102) - (N73-693))    -0.667 116.2 94
##  ((N72-3148) - (N73-1102)) - ((N73-1102) - (N73-877))   -33.556 116.2 94
##  ((N72-3148) - (N73-1102)) - ((N73-1102) - (N73-882))   -40.222 116.2 94
##  ((N72-3148) - (N73-1102)) - ((N73-1102) - (R73-81))    -63.111 116.2 94
##  ((N72-3148) - (N73-1102)) - ((N73-1102) - (R75-12))   -261.000 116.2 94
##  ((N72-3148) - (N73-1102)) - ((N73-1102) - Tracy)       -67.000 116.2 94
##  ((N72-3148) - (N73-1102)) - ((N73-693) - (N73-877))     31.667  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-693) - (N73-882))     25.000  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-693) - (R73-81))       2.111  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-693) - (R75-12))    -195.778  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-693) - Tracy)         -1.778  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-877) - (N73-882))     57.889  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-877) - (R73-81))      35.000  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-877) - (R75-12))    -162.889  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-877) - Tracy)         31.111  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-882) - (R73-81))      41.667  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-882) - (R75-12))    -156.222  94.9 94
##  ((N72-3148) - (N73-1102)) - ((N73-882) - Tracy)         37.778  94.9 94
##  ((N72-3148) - (N73-1102)) - ((R73-81) - (R75-12))     -133.333  94.9 94
##  ((N72-3148) - (N73-1102)) - ((R73-81) - Tracy)          60.667  94.9 94
##  ((N72-3148) - (N73-1102)) - ((R75-12) - Tracy)         258.556  94.9 94
##  ((N72-3148) - (N73-693)) - ((N72-3148) - (N73-877))    -32.889  67.1 94
##  ((N72-3148) - (N73-693)) - ((N72-3148) - (N73-882))    -39.556  67.1 94
##  ((N72-3148) - (N73-693)) - ((N72-3148) - (R73-81))     -62.444  67.1 94
##  ((N72-3148) - (N73-693)) - ((N72-3148) - (R75-12))    -260.333  67.1 94
##  ((N72-3148) - (N73-693)) - ((N72-3148) - Tracy)        -66.333  67.1 94
##  ((N72-3148) - (N73-693)) - ((N73-1102) - (N73-693))     64.556  67.1 94
##  ((N72-3148) - (N73-693)) - ((N73-1102) - (N73-877))     31.667  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-1102) - (N73-882))     25.000  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-1102) - (R73-81))       2.111  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-1102) - (R75-12))    -195.778  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-1102) - Tracy)         -1.778  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-693) - (N73-877))      96.889 116.2 94
##  ((N72-3148) - (N73-693)) - ((N73-693) - (N73-882))      90.222 116.2 94
##  ((N72-3148) - (N73-693)) - ((N73-693) - (R73-81))       67.333 116.2 94
##  ((N72-3148) - (N73-693)) - ((N73-693) - (R75-12))     -130.556 116.2 94
##  ((N72-3148) - (N73-693)) - ((N73-693) - Tracy)          63.444 116.2 94
##  ((N72-3148) - (N73-693)) - ((N73-877) - (N73-882))     123.111  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-877) - (R73-81))      100.222  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-877) - (R75-12))      -97.667  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-877) - Tracy)          96.333  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-882) - (R73-81))      106.889  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-882) - (R75-12))      -91.000  94.9 94
##  ((N72-3148) - (N73-693)) - ((N73-882) - Tracy)         103.000  94.9 94
##  ((N72-3148) - (N73-693)) - ((R73-81) - (R75-12))       -68.111  94.9 94
##  ((N72-3148) - (N73-693)) - ((R73-81) - Tracy)          125.889  94.9 94
##  ((N72-3148) - (N73-693)) - ((R75-12) - Tracy)          323.778  94.9 94
##  ((N72-3148) - (N73-877)) - ((N72-3148) - (N73-882))     -6.667  67.1 94
##  ((N72-3148) - (N73-877)) - ((N72-3148) - (R73-81))     -29.556  67.1 94
##  ((N72-3148) - (N73-877)) - ((N72-3148) - (R75-12))    -227.444  67.1 94
##  ((N72-3148) - (N73-877)) - ((N72-3148) - Tracy)        -33.444  67.1 94
##  ((N72-3148) - (N73-877)) - ((N73-1102) - (N73-693))     97.444  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-1102) - (N73-877))     64.556  67.1 94
##  ((N72-3148) - (N73-877)) - ((N73-1102) - (N73-882))     57.889  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-1102) - (R73-81))      35.000  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-1102) - (R75-12))    -162.889  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-1102) - Tracy)         31.111  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-693) - (N73-877))     129.778  67.1 94
##  ((N72-3148) - (N73-877)) - ((N73-693) - (N73-882))     123.111  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-693) - (R73-81))      100.222  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-693) - (R75-12))      -97.667  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-693) - Tracy)          96.333  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-877) - (N73-882))     156.000 116.2 94
##  ((N72-3148) - (N73-877)) - ((N73-877) - (R73-81))      133.111 116.2 94
##  ((N72-3148) - (N73-877)) - ((N73-877) - (R75-12))      -64.778 116.2 94
##  ((N72-3148) - (N73-877)) - ((N73-877) - Tracy)         129.222 116.2 94
##  ((N72-3148) - (N73-877)) - ((N73-882) - (R73-81))      139.778  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-882) - (R75-12))      -58.111  94.9 94
##  ((N72-3148) - (N73-877)) - ((N73-882) - Tracy)         135.889  94.9 94
##  ((N72-3148) - (N73-877)) - ((R73-81) - (R75-12))       -35.222  94.9 94
##  ((N72-3148) - (N73-877)) - ((R73-81) - Tracy)          158.778  94.9 94
##  ((N72-3148) - (N73-877)) - ((R75-12) - Tracy)          356.667  94.9 94
##  ((N72-3148) - (N73-882)) - ((N72-3148) - (R73-81))     -22.889  67.1 94
##  ((N72-3148) - (N73-882)) - ((N72-3148) - (R75-12))    -220.778  67.1 94
##  ((N72-3148) - (N73-882)) - ((N72-3148) - Tracy)        -26.778  67.1 94
##  ((N72-3148) - (N73-882)) - ((N73-1102) - (N73-693))    104.111  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-1102) - (N73-877))     71.222  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-1102) - (N73-882))     64.556  67.1 94
##  ((N72-3148) - (N73-882)) - ((N73-1102) - (R73-81))      41.667  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-1102) - (R75-12))    -156.222  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-1102) - Tracy)         37.778  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-693) - (N73-877))     136.444  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-693) - (N73-882))     129.778  67.1 94
##  ((N72-3148) - (N73-882)) - ((N73-693) - (R73-81))      106.889  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-693) - (R75-12))      -91.000  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-693) - Tracy)         103.000  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-877) - (N73-882))     162.667  67.1 94
##  ((N72-3148) - (N73-882)) - ((N73-877) - (R73-81))      139.778  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-877) - (R75-12))      -58.111  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-877) - Tracy)         135.889  94.9 94
##  ((N72-3148) - (N73-882)) - ((N73-882) - (R73-81))      146.444 116.2 94
##  ((N72-3148) - (N73-882)) - ((N73-882) - (R75-12))      -51.444 116.2 94
##  ((N72-3148) - (N73-882)) - ((N73-882) - Tracy)         142.556 116.2 94
##  ((N72-3148) - (N73-882)) - ((R73-81) - (R75-12))       -28.556  94.9 94
##  ((N72-3148) - (N73-882)) - ((R73-81) - Tracy)          165.444  94.9 94
##  ((N72-3148) - (N73-882)) - ((R75-12) - Tracy)          363.333  94.9 94
##  ((N72-3148) - (R73-81)) - ((N72-3148) - (R75-12))     -197.889  67.1 94
##  ((N72-3148) - (R73-81)) - ((N72-3148) - Tracy)          -3.889  67.1 94
##  ((N72-3148) - (R73-81)) - ((N73-1102) - (N73-693))     127.000  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-1102) - (N73-877))      94.111  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-1102) - (N73-882))      87.444  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-1102) - (R73-81))       64.556  67.1 94
##  ((N72-3148) - (R73-81)) - ((N73-1102) - (R75-12))     -133.333  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-1102) - Tracy)          60.667  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-693) - (N73-877))      159.333  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-693) - (N73-882))      152.667  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-693) - (R73-81))       129.778  67.1 94
##  ((N72-3148) - (R73-81)) - ((N73-693) - (R75-12))       -68.111  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-693) - Tracy)          125.889  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-877) - (N73-882))      185.556  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-877) - (R73-81))       162.667  67.1 94
##  ((N72-3148) - (R73-81)) - ((N73-877) - (R75-12))       -35.222  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-877) - Tracy)          158.778  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-882) - (R73-81))       169.333  67.1 94
##  ((N72-3148) - (R73-81)) - ((N73-882) - (R75-12))       -28.556  94.9 94
##  ((N72-3148) - (R73-81)) - ((N73-882) - Tracy)          165.444  94.9 94
##  ((N72-3148) - (R73-81)) - ((R73-81) - (R75-12))         -5.667 116.2 94
##  ((N72-3148) - (R73-81)) - ((R73-81) - Tracy)           188.333 116.2 94
##  ((N72-3148) - (R73-81)) - ((R75-12) - Tracy)           386.222  94.9 94
##  ((N72-3148) - (R75-12)) - ((N72-3148) - Tracy)         194.000  67.1 94
##  ((N72-3148) - (R75-12)) - ((N73-1102) - (N73-693))     324.889  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-1102) - (N73-877))     292.000  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-1102) - (N73-882))     285.333  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-1102) - (R73-81))      262.444  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-1102) - (R75-12))       64.556  67.1 94
##  ((N72-3148) - (R75-12)) - ((N73-1102) - Tracy)         258.556  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-693) - (N73-877))      357.222  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-693) - (N73-882))      350.556  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-693) - (R73-81))       327.667  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-693) - (R75-12))       129.778  67.1 94
##  ((N72-3148) - (R75-12)) - ((N73-693) - Tracy)          323.778  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-877) - (N73-882))      383.444  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-877) - (R73-81))       360.556  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-877) - (R75-12))       162.667  67.1 94
##  ((N72-3148) - (R75-12)) - ((N73-877) - Tracy)          356.667  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-882) - (R73-81))       367.222  94.9 94
##  ((N72-3148) - (R75-12)) - ((N73-882) - (R75-12))       169.333  67.1 94
##  ((N72-3148) - (R75-12)) - ((N73-882) - Tracy)          363.333  94.9 94
##  ((N72-3148) - (R75-12)) - ((R73-81) - (R75-12))        192.222  67.1 94
##  ((N72-3148) - (R75-12)) - ((R73-81) - Tracy)           386.222  94.9 94
##  ((N72-3148) - (R75-12)) - ((R75-12) - Tracy)           584.111 116.2 94
##  ((N72-3148) - Tracy) - ((N73-1102) - (N73-693))        130.889  94.9 94
##  ((N72-3148) - Tracy) - ((N73-1102) - (N73-877))         98.000  94.9 94
##  ((N72-3148) - Tracy) - ((N73-1102) - (N73-882))         91.333  94.9 94
##  ((N72-3148) - Tracy) - ((N73-1102) - (R73-81))          68.444  94.9 94
##  ((N72-3148) - Tracy) - ((N73-1102) - (R75-12))        -129.444  94.9 94
##  ((N72-3148) - Tracy) - ((N73-1102) - Tracy)             64.556  67.1 94
##  ((N72-3148) - Tracy) - ((N73-693) - (N73-877))         163.222  94.9 94
##  ((N72-3148) - Tracy) - ((N73-693) - (N73-882))         156.556  94.9 94
##  ((N72-3148) - Tracy) - ((N73-693) - (R73-81))          133.667  94.9 94
##  ((N72-3148) - Tracy) - ((N73-693) - (R75-12))          -64.222  94.9 94
##  ((N72-3148) - Tracy) - ((N73-693) - Tracy)             129.778  67.1 94
##  ((N72-3148) - Tracy) - ((N73-877) - (N73-882))         189.444  94.9 94
##  ((N72-3148) - Tracy) - ((N73-877) - (R73-81))          166.556  94.9 94
##  ((N72-3148) - Tracy) - ((N73-877) - (R75-12))          -31.333  94.9 94
##  ((N72-3148) - Tracy) - ((N73-877) - Tracy)             162.667  67.1 94
##  ((N72-3148) - Tracy) - ((N73-882) - (R73-81))          173.222  94.9 94
##  ((N72-3148) - Tracy) - ((N73-882) - (R75-12))          -24.667  94.9 94
##  ((N72-3148) - Tracy) - ((N73-882) - Tracy)             169.333  67.1 94
##  ((N72-3148) - Tracy) - ((R73-81) - (R75-12))            -1.778  94.9 94
##  ((N72-3148) - Tracy) - ((R73-81) - Tracy)              192.222  67.1 94
##  ((N72-3148) - Tracy) - ((R75-12) - Tracy)              390.111  67.1 94
##  ((N73-1102) - (N73-693)) - ((N73-1102) - (N73-877))    -32.889  67.1 94
##  ((N73-1102) - (N73-693)) - ((N73-1102) - (N73-882))    -39.556  67.1 94
##  ((N73-1102) - (N73-693)) - ((N73-1102) - (R73-81))     -62.444  67.1 94
##  ((N73-1102) - (N73-693)) - ((N73-1102) - (R75-12))    -260.333  67.1 94
##  ((N73-1102) - (N73-693)) - ((N73-1102) - Tracy)        -66.333  67.1 94
##  ((N73-1102) - (N73-693)) - ((N73-693) - (N73-877))      32.333 116.2 94
##  ((N73-1102) - (N73-693)) - ((N73-693) - (N73-882))      25.667 116.2 94
##  ((N73-1102) - (N73-693)) - ((N73-693) - (R73-81))        2.778 116.2 94
##  ((N73-1102) - (N73-693)) - ((N73-693) - (R75-12))     -195.111 116.2 94
##  ((N73-1102) - (N73-693)) - ((N73-693) - Tracy)          -1.111 116.2 94
##  ((N73-1102) - (N73-693)) - ((N73-877) - (N73-882))      58.556  94.9 94
##  ((N73-1102) - (N73-693)) - ((N73-877) - (R73-81))       35.667  94.9 94
##  ((N73-1102) - (N73-693)) - ((N73-877) - (R75-12))     -162.222  94.9 94
##  ((N73-1102) - (N73-693)) - ((N73-877) - Tracy)          31.778  94.9 94
##  ((N73-1102) - (N73-693)) - ((N73-882) - (R73-81))       42.333  94.9 94
##  ((N73-1102) - (N73-693)) - ((N73-882) - (R75-12))     -155.556  94.9 94
##  ((N73-1102) - (N73-693)) - ((N73-882) - Tracy)          38.444  94.9 94
##  ((N73-1102) - (N73-693)) - ((R73-81) - (R75-12))      -132.667  94.9 94
##  ((N73-1102) - (N73-693)) - ((R73-81) - Tracy)           61.333  94.9 94
##  ((N73-1102) - (N73-693)) - ((R75-12) - Tracy)          259.222  94.9 94
##  ((N73-1102) - (N73-877)) - ((N73-1102) - (N73-882))     -6.667  67.1 94
##  ((N73-1102) - (N73-877)) - ((N73-1102) - (R73-81))     -29.556  67.1 94
##  ((N73-1102) - (N73-877)) - ((N73-1102) - (R75-12))    -227.444  67.1 94
##  ((N73-1102) - (N73-877)) - ((N73-1102) - Tracy)        -33.444  67.1 94
##  ((N73-1102) - (N73-877)) - ((N73-693) - (N73-877))      65.222  67.1 94
##  ((N73-1102) - (N73-877)) - ((N73-693) - (N73-882))      58.556  94.9 94
##  ((N73-1102) - (N73-877)) - ((N73-693) - (R73-81))       35.667  94.9 94
##  ((N73-1102) - (N73-877)) - ((N73-693) - (R75-12))     -162.222  94.9 94
##  ((N73-1102) - (N73-877)) - ((N73-693) - Tracy)          31.778  94.9 94
##  ((N73-1102) - (N73-877)) - ((N73-877) - (N73-882))      91.444 116.2 94
##  ((N73-1102) - (N73-877)) - ((N73-877) - (R73-81))       68.556 116.2 94
##  ((N73-1102) - (N73-877)) - ((N73-877) - (R75-12))     -129.333 116.2 94
##  ((N73-1102) - (N73-877)) - ((N73-877) - Tracy)          64.667 116.2 94
##  ((N73-1102) - (N73-877)) - ((N73-882) - (R73-81))       75.222  94.9 94
##  ((N73-1102) - (N73-877)) - ((N73-882) - (R75-12))     -122.667  94.9 94
##  ((N73-1102) - (N73-877)) - ((N73-882) - Tracy)          71.333  94.9 94
##  ((N73-1102) - (N73-877)) - ((R73-81) - (R75-12))       -99.778  94.9 94
##  ((N73-1102) - (N73-877)) - ((R73-81) - Tracy)           94.222  94.9 94
##  ((N73-1102) - (N73-877)) - ((R75-12) - Tracy)          292.111  94.9 94
##  ((N73-1102) - (N73-882)) - ((N73-1102) - (R73-81))     -22.889  67.1 94
##  ((N73-1102) - (N73-882)) - ((N73-1102) - (R75-12))    -220.778  67.1 94
##  ((N73-1102) - (N73-882)) - ((N73-1102) - Tracy)        -26.778  67.1 94
##  ((N73-1102) - (N73-882)) - ((N73-693) - (N73-877))      71.889  94.9 94
##  ((N73-1102) - (N73-882)) - ((N73-693) - (N73-882))      65.222  67.1 94
##  ((N73-1102) - (N73-882)) - ((N73-693) - (R73-81))       42.333  94.9 94
##  ((N73-1102) - (N73-882)) - ((N73-693) - (R75-12))     -155.556  94.9 94
##  ((N73-1102) - (N73-882)) - ((N73-693) - Tracy)          38.444  94.9 94
##  ((N73-1102) - (N73-882)) - ((N73-877) - (N73-882))      98.111  67.1 94
##  ((N73-1102) - (N73-882)) - ((N73-877) - (R73-81))       75.222  94.9 94
##  ((N73-1102) - (N73-882)) - ((N73-877) - (R75-12))     -122.667  94.9 94
##  ((N73-1102) - (N73-882)) - ((N73-877) - Tracy)          71.333  94.9 94
##  ((N73-1102) - (N73-882)) - ((N73-882) - (R73-81))       81.889 116.2 94
##  ((N73-1102) - (N73-882)) - ((N73-882) - (R75-12))     -116.000 116.2 94
##  ((N73-1102) - (N73-882)) - ((N73-882) - Tracy)          78.000 116.2 94
##  ((N73-1102) - (N73-882)) - ((R73-81) - (R75-12))       -93.111  94.9 94
##  ((N73-1102) - (N73-882)) - ((R73-81) - Tracy)          100.889  94.9 94
##  ((N73-1102) - (N73-882)) - ((R75-12) - Tracy)          298.778  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-1102) - (R75-12))     -197.889  67.1 94
##  ((N73-1102) - (R73-81)) - ((N73-1102) - Tracy)          -3.889  67.1 94
##  ((N73-1102) - (R73-81)) - ((N73-693) - (N73-877))       94.778  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-693) - (N73-882))       88.111  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-693) - (R73-81))        65.222  67.1 94
##  ((N73-1102) - (R73-81)) - ((N73-693) - (R75-12))      -132.667  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-693) - Tracy)           61.333  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-877) - (N73-882))      121.000  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-877) - (R73-81))        98.111  67.1 94
##  ((N73-1102) - (R73-81)) - ((N73-877) - (R75-12))       -99.778  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-877) - Tracy)           94.222  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-882) - (R73-81))       104.778  67.1 94
##  ((N73-1102) - (R73-81)) - ((N73-882) - (R75-12))       -93.111  94.9 94
##  ((N73-1102) - (R73-81)) - ((N73-882) - Tracy)          100.889  94.9 94
##  ((N73-1102) - (R73-81)) - ((R73-81) - (R75-12))        -70.222 116.2 94
##  ((N73-1102) - (R73-81)) - ((R73-81) - Tracy)           123.778 116.2 94
##  ((N73-1102) - (R73-81)) - ((R75-12) - Tracy)           321.667  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-1102) - Tracy)         194.000  67.1 94
##  ((N73-1102) - (R75-12)) - ((N73-693) - (N73-877))      292.667  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-693) - (N73-882))      286.000  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-693) - (R73-81))       263.111  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-693) - (R75-12))        65.222  67.1 94
##  ((N73-1102) - (R75-12)) - ((N73-693) - Tracy)          259.222  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-877) - (N73-882))      318.889  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-877) - (R73-81))       296.000  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-877) - (R75-12))        98.111  67.1 94
##  ((N73-1102) - (R75-12)) - ((N73-877) - Tracy)          292.111  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-882) - (R73-81))       302.667  94.9 94
##  ((N73-1102) - (R75-12)) - ((N73-882) - (R75-12))       104.778  67.1 94
##  ((N73-1102) - (R75-12)) - ((N73-882) - Tracy)          298.778  94.9 94
##  ((N73-1102) - (R75-12)) - ((R73-81) - (R75-12))        127.667  67.1 94
##  ((N73-1102) - (R75-12)) - ((R73-81) - Tracy)           321.667  94.9 94
##  ((N73-1102) - (R75-12)) - ((R75-12) - Tracy)           519.556 116.2 94
##  ((N73-1102) - Tracy) - ((N73-693) - (N73-877))          98.667  94.9 94
##  ((N73-1102) - Tracy) - ((N73-693) - (N73-882))          92.000  94.9 94
##  ((N73-1102) - Tracy) - ((N73-693) - (R73-81))           69.111  94.9 94
##  ((N73-1102) - Tracy) - ((N73-693) - (R75-12))         -128.778  94.9 94
##  ((N73-1102) - Tracy) - ((N73-693) - Tracy)              65.222  67.1 94
##  ((N73-1102) - Tracy) - ((N73-877) - (N73-882))         124.889  94.9 94
##  ((N73-1102) - Tracy) - ((N73-877) - (R73-81))          102.000  94.9 94
##  ((N73-1102) - Tracy) - ((N73-877) - (R75-12))          -95.889  94.9 94
##  ((N73-1102) - Tracy) - ((N73-877) - Tracy)              98.111  67.1 94
##  ((N73-1102) - Tracy) - ((N73-882) - (R73-81))          108.667  94.9 94
##  ((N73-1102) - Tracy) - ((N73-882) - (R75-12))          -89.222  94.9 94
##  ((N73-1102) - Tracy) - ((N73-882) - Tracy)             104.778  67.1 94
##  ((N73-1102) - Tracy) - ((R73-81) - (R75-12))           -66.333  94.9 94
##  ((N73-1102) - Tracy) - ((R73-81) - Tracy)              127.667  67.1 94
##  ((N73-1102) - Tracy) - ((R75-12) - Tracy)              325.556  67.1 94
##  ((N73-693) - (N73-877)) - ((N73-693) - (N73-882))       -6.667  67.1 94
##  ((N73-693) - (N73-877)) - ((N73-693) - (R73-81))       -29.556  67.1 94
##  ((N73-693) - (N73-877)) - ((N73-693) - (R75-12))      -227.444  67.1 94
##  ((N73-693) - (N73-877)) - ((N73-693) - Tracy)          -33.444  67.1 94
##  ((N73-693) - (N73-877)) - ((N73-877) - (N73-882))       26.222 116.2 94
##  ((N73-693) - (N73-877)) - ((N73-877) - (R73-81))         3.333 116.2 94
##  ((N73-693) - (N73-877)) - ((N73-877) - (R75-12))      -194.556 116.2 94
##  ((N73-693) - (N73-877)) - ((N73-877) - Tracy)           -0.556 116.2 94
##  ((N73-693) - (N73-877)) - ((N73-882) - (R73-81))        10.000  94.9 94
##  ((N73-693) - (N73-877)) - ((N73-882) - (R75-12))      -187.889  94.9 94
##  ((N73-693) - (N73-877)) - ((N73-882) - Tracy)            6.111  94.9 94
##  ((N73-693) - (N73-877)) - ((R73-81) - (R75-12))       -165.000  94.9 94
##  ((N73-693) - (N73-877)) - ((R73-81) - Tracy)            29.000  94.9 94
##  ((N73-693) - (N73-877)) - ((R75-12) - Tracy)           226.889  94.9 94
##  ((N73-693) - (N73-882)) - ((N73-693) - (R73-81))       -22.889  67.1 94
##  ((N73-693) - (N73-882)) - ((N73-693) - (R75-12))      -220.778  67.1 94
##  ((N73-693) - (N73-882)) - ((N73-693) - Tracy)          -26.778  67.1 94
##  ((N73-693) - (N73-882)) - ((N73-877) - (N73-882))       32.889  67.1 94
##  ((N73-693) - (N73-882)) - ((N73-877) - (R73-81))        10.000  94.9 94
##  ((N73-693) - (N73-882)) - ((N73-877) - (R75-12))      -187.889  94.9 94
##  ((N73-693) - (N73-882)) - ((N73-877) - Tracy)            6.111  94.9 94
##  ((N73-693) - (N73-882)) - ((N73-882) - (R73-81))        16.667 116.2 94
##  ((N73-693) - (N73-882)) - ((N73-882) - (R75-12))      -181.222 116.2 94
##  ((N73-693) - (N73-882)) - ((N73-882) - Tracy)           12.778 116.2 94
##  ((N73-693) - (N73-882)) - ((R73-81) - (R75-12))       -158.333  94.9 94
##  ((N73-693) - (N73-882)) - ((R73-81) - Tracy)            35.667  94.9 94
##  ((N73-693) - (N73-882)) - ((R75-12) - Tracy)           233.556  94.9 94
##  ((N73-693) - (R73-81)) - ((N73-693) - (R75-12))       -197.889  67.1 94
##  ((N73-693) - (R73-81)) - ((N73-693) - Tracy)            -3.889  67.1 94
##  ((N73-693) - (R73-81)) - ((N73-877) - (N73-882))        55.778  94.9 94
##  ((N73-693) - (R73-81)) - ((N73-877) - (R73-81))         32.889  67.1 94
##  ((N73-693) - (R73-81)) - ((N73-877) - (R75-12))       -165.000  94.9 94
##  ((N73-693) - (R73-81)) - ((N73-877) - Tracy)            29.000  94.9 94
##  ((N73-693) - (R73-81)) - ((N73-882) - (R73-81))         39.556  67.1 94
##  ((N73-693) - (R73-81)) - ((N73-882) - (R75-12))       -158.333  94.9 94
##  ((N73-693) - (R73-81)) - ((N73-882) - Tracy)            35.667  94.9 94
##  ((N73-693) - (R73-81)) - ((R73-81) - (R75-12))        -135.444 116.2 94
##  ((N73-693) - (R73-81)) - ((R73-81) - Tracy)             58.556 116.2 94
##  ((N73-693) - (R73-81)) - ((R75-12) - Tracy)            256.444  94.9 94
##  ((N73-693) - (R75-12)) - ((N73-693) - Tracy)           194.000  67.1 94
##  ((N73-693) - (R75-12)) - ((N73-877) - (N73-882))       253.667  94.9 94
##  ((N73-693) - (R75-12)) - ((N73-877) - (R73-81))        230.778  94.9 94
##  ((N73-693) - (R75-12)) - ((N73-877) - (R75-12))         32.889  67.1 94
##  ((N73-693) - (R75-12)) - ((N73-877) - Tracy)           226.889  94.9 94
##  ((N73-693) - (R75-12)) - ((N73-882) - (R73-81))        237.444  94.9 94
##  ((N73-693) - (R75-12)) - ((N73-882) - (R75-12))         39.556  67.1 94
##  ((N73-693) - (R75-12)) - ((N73-882) - Tracy)           233.556  94.9 94
##  ((N73-693) - (R75-12)) - ((R73-81) - (R75-12))          62.444  67.1 94
##  ((N73-693) - (R75-12)) - ((R73-81) - Tracy)            256.444  94.9 94
##  ((N73-693) - (R75-12)) - ((R75-12) - Tracy)            454.333 116.2 94
##  ((N73-693) - Tracy) - ((N73-877) - (N73-882))           59.667  94.9 94
##  ((N73-693) - Tracy) - ((N73-877) - (R73-81))            36.778  94.9 94
##  ((N73-693) - Tracy) - ((N73-877) - (R75-12))          -161.111  94.9 94
##  ((N73-693) - Tracy) - ((N73-877) - Tracy)               32.889  67.1 94
##  ((N73-693) - Tracy) - ((N73-882) - (R73-81))            43.444  94.9 94
##  ((N73-693) - Tracy) - ((N73-882) - (R75-12))          -154.444  94.9 94
##  ((N73-693) - Tracy) - ((N73-882) - Tracy)               39.556  67.1 94
##  ((N73-693) - Tracy) - ((R73-81) - (R75-12))           -131.556  94.9 94
##  ((N73-693) - Tracy) - ((R73-81) - Tracy)                62.444  67.1 94
##  ((N73-693) - Tracy) - ((R75-12) - Tracy)               260.333  67.1 94
##  ((N73-877) - (N73-882)) - ((N73-877) - (R73-81))       -22.889  67.1 94
##  ((N73-877) - (N73-882)) - ((N73-877) - (R75-12))      -220.778  67.1 94
##  ((N73-877) - (N73-882)) - ((N73-877) - Tracy)          -26.778  67.1 94
##  ((N73-877) - (N73-882)) - ((N73-882) - (R73-81))       -16.222 116.2 94
##  ((N73-877) - (N73-882)) - ((N73-882) - (R75-12))      -214.111 116.2 94
##  ((N73-877) - (N73-882)) - ((N73-882) - Tracy)          -20.111 116.2 94
##  ((N73-877) - (N73-882)) - ((R73-81) - (R75-12))       -191.222  94.9 94
##  ((N73-877) - (N73-882)) - ((R73-81) - Tracy)             2.778  94.9 94
##  ((N73-877) - (N73-882)) - ((R75-12) - Tracy)           200.667  94.9 94
##  ((N73-877) - (R73-81)) - ((N73-877) - (R75-12))       -197.889  67.1 94
##  ((N73-877) - (R73-81)) - ((N73-877) - Tracy)            -3.889  67.1 94
##  ((N73-877) - (R73-81)) - ((N73-882) - (R73-81))          6.667  67.1 94
##  ((N73-877) - (R73-81)) - ((N73-882) - (R75-12))       -191.222  94.9 94
##  ((N73-877) - (R73-81)) - ((N73-882) - Tracy)             2.778  94.9 94
##  ((N73-877) - (R73-81)) - ((R73-81) - (R75-12))        -168.333 116.2 94
##  ((N73-877) - (R73-81)) - ((R73-81) - Tracy)             25.667 116.2 94
##  ((N73-877) - (R73-81)) - ((R75-12) - Tracy)            223.556  94.9 94
##  ((N73-877) - (R75-12)) - ((N73-877) - Tracy)           194.000  67.1 94
##  ((N73-877) - (R75-12)) - ((N73-882) - (R73-81))        204.556  94.9 94
##  ((N73-877) - (R75-12)) - ((N73-882) - (R75-12))          6.667  67.1 94
##  ((N73-877) - (R75-12)) - ((N73-882) - Tracy)           200.667  94.9 94
##  ((N73-877) - (R75-12)) - ((R73-81) - (R75-12))          29.556  67.1 94
##  ((N73-877) - (R75-12)) - ((R73-81) - Tracy)            223.556  94.9 94
##  ((N73-877) - (R75-12)) - ((R75-12) - Tracy)            421.444 116.2 94
##  ((N73-877) - Tracy) - ((N73-882) - (R73-81))            10.556  94.9 94
##  ((N73-877) - Tracy) - ((N73-882) - (R75-12))          -187.333  94.9 94
##  ((N73-877) - Tracy) - ((N73-882) - Tracy)                6.667  67.1 94
##  ((N73-877) - Tracy) - ((R73-81) - (R75-12))           -164.444  94.9 94
##  ((N73-877) - Tracy) - ((R73-81) - Tracy)                29.556  67.1 94
##  ((N73-877) - Tracy) - ((R75-12) - Tracy)               227.444  67.1 94
##  ((N73-882) - (R73-81)) - ((N73-882) - (R75-12))       -197.889  67.1 94
##  ((N73-882) - (R73-81)) - ((N73-882) - Tracy)            -3.889  67.1 94
##  ((N73-882) - (R73-81)) - ((R73-81) - (R75-12))        -175.000 116.2 94
##  ((N73-882) - (R73-81)) - ((R73-81) - Tracy)             19.000 116.2 94
##  ((N73-882) - (R73-81)) - ((R75-12) - Tracy)            216.889  94.9 94
##  ((N73-882) - (R75-12)) - ((N73-882) - Tracy)           194.000  67.1 94
##  ((N73-882) - (R75-12)) - ((R73-81) - (R75-12))          22.889  67.1 94
##  ((N73-882) - (R75-12)) - ((R73-81) - Tracy)            216.889  94.9 94
##  ((N73-882) - (R75-12)) - ((R75-12) - Tracy)            414.778 116.2 94
##  ((N73-882) - Tracy) - ((R73-81) - (R75-12))           -171.111  94.9 94
##  ((N73-882) - Tracy) - ((R73-81) - Tracy)                22.889  67.1 94
##  ((N73-882) - Tracy) - ((R75-12) - Tracy)               220.778  67.1 94
##  ((R73-81) - (R75-12)) - ((R73-81) - Tracy)             194.000  67.1 94
##  ((R73-81) - (R75-12)) - ((R75-12) - Tracy)             391.889 116.2 94
##  ((R73-81) - Tracy) - ((R75-12) - Tracy)                197.889  67.1 94
##  t.ratio p.value
##   1.155  1.0000 
##  -1.130  1.0000 
##   2.379  0.9869 
##   1.416  1.0000 
##   0.444  1.0000 
##  -0.046  1.0000 
##  -0.146  1.0000 
##  -0.487  1.0000 
##  -3.437  0.4057 
##  -0.545  1.0000 
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##   1.903  0.9999 
##   4.854  0.0069 
##  -0.099  1.0000 
##  -0.441  1.0000 
##  -3.391  0.4405 
##  -0.499  1.0000 
##   0.226  1.0000 
##   0.029  1.0000 
##  -1.675  1.0000 
##  -0.005  1.0000 
##   0.105  1.0000 
##  -1.981  0.9997 
##   0.064  1.0000 
##  -1.740  1.0000 
##   0.306  1.0000 
##   2.392  0.9856 
##  -0.341  1.0000 
##  -3.292  0.5183 
##  -0.399  1.0000 
##   0.490  1.0000 
##   0.105  1.0000 
##  -1.981  0.9997 
##   0.064  1.0000 
##   0.143  1.0000 
##  -1.560  1.0000 
##   0.110  1.0000 
##  -1.669  1.0000 
##   0.376  1.0000 
##   2.462  0.9766 
##  -2.950  0.7797 
##  -0.058  1.0000 
##   0.588  1.0000 
##   0.490  1.0000 
##  -1.740  1.0000 
##   0.306  1.0000 
##   0.590  1.0000 
##  -1.669  1.0000 
##   0.376  1.0000 
##  -1.166  1.0000 
##   0.504  1.0000 
##   2.704  0.9126 
##   2.892  0.8170 
##   2.674  0.9237 
##   2.433  0.9808 
##   0.490  1.0000 
##   2.392  0.9856 
##   2.503  0.9697 
##   0.590  1.0000 
##   2.462  0.9766 
##   0.931  1.0000 
##   2.704  0.9126 
##   3.911  0.1407 
##   0.629  1.0000 
##   0.388  1.0000 
##  -1.699  1.0000 
##   0.490  1.0000 
##   0.458  1.0000 
##  -1.628  1.0000 
##   0.590  1.0000 
##  -1.387  1.0000 
##   0.931  1.0000 
##   3.881  0.1519 
##  -0.341  1.0000 
##  -3.292  0.5183 
##  -0.399  1.0000 
##  -0.140  1.0000 
##  -1.843  1.0000 
##  -0.173  1.0000 
##  -2.016  0.9996 
##   0.029  1.0000 
##   2.116  0.9987 
##  -2.950  0.7797 
##  -0.058  1.0000 
##   0.099  1.0000 
##  -2.016  0.9996 
##   0.029  1.0000 
##  -1.449  1.0000 
##   0.221  1.0000 
##   2.357  0.9889 
##   2.892  0.8170 
##   2.157  0.9981 
##   0.099  1.0000 
##   2.116  0.9987 
##   0.441  1.0000 
##   2.357  0.9889 
##   3.628  0.2774 
##   0.111  1.0000 
##  -1.975  0.9998 
##   0.099  1.0000 
##  -1.734  1.0000 
##   0.441  1.0000 
##   3.391  0.4405 
##  -2.950  0.7797 
##  -0.058  1.0000 
##  -1.506  1.0000 
##   0.164  1.0000 
##   2.287  0.9936 
##   2.892  0.8170 
##   0.341  1.0000 
##   2.287  0.9936 
##   3.570  0.3133 
##  -1.804  1.0000 
##   0.341  1.0000 
##   3.292  0.5183 
##   2.892  0.8170 
##   3.373  0.4541 
##   2.950  0.7797 
## 
## Results are averaged over the levels of: loc 
## P value adjustment: tukey method for comparing a family of 66 estimates
library(agricolae)
HSD.test(soy.lm.n, "gen", group = TRUE, console = TRUE)
## 
## Study: soy.lm.n ~ "gen"
## 
## HSD Test for yield 
## 
## Mean Square Error:  20243.4 
## 
## gen,  means
## 
##               yield      std r  Min  Max
## Centennial 1394.778 200.4631 9 1180 1713
## D74-7741   1406.444 301.6074 9 1099 1851
## N72-137    1483.889 188.6528 9 1177 1723
## N72-3058   1330.667 241.7861 9  990 1647
## N72-3148   1566.000 221.5835 9 1303 1929
## N73-1102   1501.444 228.3808 9 1178 1894
## N73-693    1436.222 163.4134 9 1214 1642
## N73-877    1403.333 236.8444 9 1107 1775
## N73-882    1396.667 191.5548 9 1096 1673
## R73-81     1373.778 271.7930 9 1064 1800
## R75-12     1175.889 181.7852 9  911 1422
## Tracy      1369.889 272.3850 9  960 1841
## 
## Alpha: 0.05 ; DF Error: 94 
## Critical Value of Studentized Range: 4.740283 
## 
## Minimun Significant Difference: 224.8147 
## 
## Treatments with the same letter are not significantly different.
## 
##               yield groups
## N72-3148   1566.000      a
## N73-1102   1501.444     ab
## N72-137    1483.889     ab
## N73-693    1436.222     ab
## D74-7741   1406.444     ab
## N73-877    1403.333     ab
## N73-882    1396.667    abc
## Centennial 1394.778    abc
## R73-81     1373.778    abc
## Tracy      1369.889    abc
## N72-3058   1330.667     bc
## R75-12     1175.889      c

The only significant difference is between (N72-3058) and (N72-3148) (p-value = 0.0318).

f.) According to the model from e.), how many kg of yield do we obtain for the Centennial genotype, grown in Plymouth, on average? Justify your answer.

coef(summary(soy.lm.n))
##                Estimate Std. Error     t value     Pr(>|t|)
## (Intercept) 1193.583333   51.22641 23.30015787 7.831812e-41
## genD74-7741   11.666667   67.07111  0.17394474 8.622829e-01
## genN72-137    89.111111   67.07111  1.32860650 1.871942e-01
## genN72-3058  -64.111111   67.07111 -0.95586777 3.415909e-01
## genN72-3148  171.222222   67.07111  2.55284616 1.229518e-02
## genN73-1102  106.666667   67.07111  1.59035192 1.151132e-01
## genN73-693    41.444444   67.07111  0.61791799 5.381236e-01
## genN73-877     8.555556   67.07111  0.12755948 8.987701e-01
## genN73-882     1.888889   67.07111  0.02816248 9.775923e-01
## genR73-81    -21.000000   67.07111 -0.31310053 7.548979e-01
## genR75-12   -218.888889   67.07111 -3.26353467 1.534889e-03
## genTracy     -24.888889   67.07111 -0.37108212 7.114114e-01
## locClinton   415.305556   33.53555 12.38403729 1.842719e-21
## locPlymouth  188.277778   33.53555  5.61427361 1.991500e-07
coef(summary(soy.lm.n))['(Intercept)', 'Estimate'] + coef(summary(soy.lm.n))['locPlymouth', 'Estimate']
## [1] 1381.861
#or 

pred.df = data.frame(gen = "Centennial", loc = "Plymouth")
predict(soy.lm.n, pred.df)
##        1 
## 1381.861

The average yield for the genotype ‘Centennial’ (reference level) and the location ‘Plymouth’ is 1381.861 kg

g.) How high is the variance of the yield values, and what proportion of it is explained with the model from e.)? Explain your approach.

summary(soy.lm.n)
## 
## Call:
## lm(formula = yield ~ gen + loc, data = soy)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -329.00  -99.49  -13.31   82.08  440.31 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1193.583     51.226  23.300  < 2e-16 ***
## genD74-7741   11.667     67.071   0.174  0.86228    
## genN72-137    89.111     67.071   1.329  0.18719    
## genN72-3058  -64.111     67.071  -0.956  0.34159    
## genN72-3148  171.222     67.071   2.553  0.01230 *  
## genN73-1102  106.667     67.071   1.590  0.11511    
## genN73-693    41.444     67.071   0.618  0.53812    
## genN73-877     8.556     67.071   0.128  0.89877    
## genN73-882     1.889     67.071   0.028  0.97759    
## genR73-81    -21.000     67.071  -0.313  0.75490    
## genR75-12   -218.889     67.071  -3.264  0.00153 ** 
## genTracy     -24.889     67.071  -0.371  0.71141    
## locClinton   415.306     33.536  12.384  < 2e-16 ***
## locPlymouth  188.278     33.536   5.614 1.99e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 142.3 on 94 degrees of freedom
## Multiple R-squared:  0.6797, Adjusted R-squared:  0.6354 
## F-statistic: 15.35 on 13 and 94 DF,  p-value: < 2.2e-16
yield.var = var(soy$yield)
yield.var
## [1] 55528.94
r.squared = var(fitted(soy.lm.n))/yield.var
r.squared
## [1] 0.6797359

The variance of the yield variable is 55528.94 and 67.97% is explained by the model.

h.) Perform the required model diagnostics (assumption checks), report your R code and the results.

library(car)
qqPlot(resid(soy.lm.n))

## [1] 63 75
plot(soy.lm.n, which=1)

Accounting for block effects

b.) Do you find evidence of an effect of the location, the genotype or their interaction on the average yield? First, give only your R code to answer this question, . . .

c.) . . . then name the tools you use, give p values for each effect and answer the question.

soy.lme = lme(yield~gen*loc, random=~1|block, data=soy)

anova(soy.lme, type='marginal')
summary(soy.lme)
## Linear mixed-effects model fit by REML
##  Data: soy 
##        AIC      BIC    logLik
##   1029.339 1115.853 -476.6696
## 
## Random effects:
##  Formula: ~1 | block
##         (Intercept) Residual
## StdDev:  0.01037198 137.9402
## 
## Fixed effects: yield ~ gen * loc 
##                             Value Std.Error DF   t-value p-value
## (Intercept)             1200.6667  79.63979 70 15.076215  0.0000
## genD74-7741              -93.6667 112.62768 70 -0.831649  0.4084
## genN72-137               249.6667 112.62768 70  2.216743  0.0299
## genN72-3058              -94.0000 112.62768 70 -0.834608  0.4068
## genN72-3148              127.0000 112.62768 70  1.127609  0.2633
## genN73-1102               62.0000 112.62768 70  0.550486  0.5837
## genN73-693                74.0000 112.62768 70  0.657032  0.5133
## genN73-877                12.0000 112.62768 70  0.106546  0.9155
## genN73-882               -26.6667 112.62768 70 -0.236768  0.8135
## genR73-81                -68.3333 112.62768 70 -0.606719  0.5460
## genR75-12               -100.3333 112.62768 70 -0.890841  0.3761
## genTracy                -125.0000 112.62768 70 -1.109852  0.2709
## locClinton               391.0000 112.62768 70  3.471616  0.0009
## locPlymouth              191.3333 112.62768 70  1.698813  0.0938
## genD74-7741:locClinton   232.6667 159.27959 70  1.460744  0.1486
## genN72-137:locClinton   -300.0000 159.27959 70 -1.883480  0.0638
## genN72-3058:locClinton   101.3333 159.27959 70  0.636198  0.5267
## genN72-3148:locClinton    65.3333 159.27959 70  0.410180  0.6829
## genN73-1102:locClinton    77.0000 159.27959 70  0.483427  0.6303
## genN73-693:locClinton    -94.6667 159.27959 70 -0.594343  0.5542
## genN73-877:locClinton     15.6667 159.27959 70  0.098360  0.9219
## genN73-882:locClinton    -41.6667 159.27959 70 -0.261595  0.7944
## genR73-81:locClinton     179.0000 159.27959 70  1.123810  0.2649
## genR75-12:locClinton    -105.6667 159.27959 70 -0.663404  0.5093
## genTracy:locClinton      162.6667 159.27959 70  1.021265  0.3106
## genD74-7741:locPlymouth   83.3333 159.27959 70  0.523189  0.6025
## genN72-137:locPlymouth  -181.6667 159.27959 70 -1.140552  0.2579
## genN72-3058:locPlymouth  -11.6667 159.27959 70 -0.073246  0.9418
## genN72-3148:locPlymouth   67.3333 159.27959 70  0.422737  0.6738
## genN73-1102:locPlymouth   57.0000 159.27959 70  0.357861  0.7215
## genN73-693:locPlymouth    -3.0000 159.27959 70 -0.018835  0.9850
## genN73-877:locPlymouth   -26.0000 159.27959 70 -0.163235  0.8708
## genN73-882:locPlymouth   127.3333 159.27959 70  0.799433  0.4267
## genR73-81:locPlymouth    -37.0000 159.27959 70 -0.232296  0.8170
## genR75-12:locPlymouth   -250.0000 159.27959 70 -1.569567  0.1210
## genTracy:locPlymouth     137.6667 159.27959 70  0.864308  0.3904
##  Correlation: 
##                         (Intr) gnD74-7741 gnN72-137 gnN72-3058 gnN72-3148
## genD74-7741             -0.707                                           
## genN72-137              -0.707  0.500                                    
## genN72-3058             -0.707  0.500      0.500                         
## genN72-3148             -0.707  0.500      0.500     0.500               
## genN73-1102             -0.707  0.500      0.500     0.500      0.500    
## genN73-693              -0.707  0.500      0.500     0.500      0.500    
## genN73-877              -0.707  0.500      0.500     0.500      0.500    
## genN73-882              -0.707  0.500      0.500     0.500      0.500    
## genR73-81               -0.707  0.500      0.500     0.500      0.500    
## genR75-12               -0.707  0.500      0.500     0.500      0.500    
## genTracy                -0.707  0.500      0.500     0.500      0.500    
## locClinton              -0.707  0.500      0.500     0.500      0.500    
## locPlymouth             -0.707  0.500      0.500     0.500      0.500    
## genD74-7741:locClinton   0.500 -0.707     -0.354    -0.354     -0.354    
## genN72-137:locClinton    0.500 -0.354     -0.707    -0.354     -0.354    
## genN72-3058:locClinton   0.500 -0.354     -0.354    -0.707     -0.354    
## genN72-3148:locClinton   0.500 -0.354     -0.354    -0.354     -0.707    
## genN73-1102:locClinton   0.500 -0.354     -0.354    -0.354     -0.354    
## genN73-693:locClinton    0.500 -0.354     -0.354    -0.354     -0.354    
## genN73-877:locClinton    0.500 -0.354     -0.354    -0.354     -0.354    
## genN73-882:locClinton    0.500 -0.354     -0.354    -0.354     -0.354    
## genR73-81:locClinton     0.500 -0.354     -0.354    -0.354     -0.354    
## genR75-12:locClinton     0.500 -0.354     -0.354    -0.354     -0.354    
## genTracy:locClinton      0.500 -0.354     -0.354    -0.354     -0.354    
## genD74-7741:locPlymouth  0.500 -0.707     -0.354    -0.354     -0.354    
## genN72-137:locPlymouth   0.500 -0.354     -0.707    -0.354     -0.354    
## genN72-3058:locPlymouth  0.500 -0.354     -0.354    -0.707     -0.354    
## genN72-3148:locPlymouth  0.500 -0.354     -0.354    -0.354     -0.707    
## genN73-1102:locPlymouth  0.500 -0.354     -0.354    -0.354     -0.354    
## genN73-693:locPlymouth   0.500 -0.354     -0.354    -0.354     -0.354    
## genN73-877:locPlymouth   0.500 -0.354     -0.354    -0.354     -0.354    
## genN73-882:locPlymouth   0.500 -0.354     -0.354    -0.354     -0.354    
## genR73-81:locPlymouth    0.500 -0.354     -0.354    -0.354     -0.354    
## genR75-12:locPlymouth    0.500 -0.354     -0.354    -0.354     -0.354    
## genTracy:locPlymouth     0.500 -0.354     -0.354    -0.354     -0.354    
##                         gnN73-1102 gnN73-693 gnN73-877 gnN73-882 gnR73-81
## genD74-7741                                                              
## genN72-137                                                               
## genN72-3058                                                              
## genN72-3148                                                              
## genN73-1102                                                              
## genN73-693               0.500                                           
## genN73-877               0.500      0.500                                
## genN73-882               0.500      0.500     0.500                      
## genR73-81                0.500      0.500     0.500     0.500            
## genR75-12                0.500      0.500     0.500     0.500     0.500  
## genTracy                 0.500      0.500     0.500     0.500     0.500  
## locClinton               0.500      0.500     0.500     0.500     0.500  
## locPlymouth              0.500      0.500     0.500     0.500     0.500  
## genD74-7741:locClinton  -0.354     -0.354    -0.354    -0.354    -0.354  
## genN72-137:locClinton   -0.354     -0.354    -0.354    -0.354    -0.354  
## genN72-3058:locClinton  -0.354     -0.354    -0.354    -0.354    -0.354  
## genN72-3148:locClinton  -0.354     -0.354    -0.354    -0.354    -0.354  
## genN73-1102:locClinton  -0.707     -0.354    -0.354    -0.354    -0.354  
## genN73-693:locClinton   -0.354     -0.707    -0.354    -0.354    -0.354  
## genN73-877:locClinton   -0.354     -0.354    -0.707    -0.354    -0.354  
## genN73-882:locClinton   -0.354     -0.354    -0.354    -0.707    -0.354  
## genR73-81:locClinton    -0.354     -0.354    -0.354    -0.354    -0.707  
## genR75-12:locClinton    -0.354     -0.354    -0.354    -0.354    -0.354  
## genTracy:locClinton     -0.354     -0.354    -0.354    -0.354    -0.354  
## genD74-7741:locPlymouth -0.354     -0.354    -0.354    -0.354    -0.354  
## genN72-137:locPlymouth  -0.354     -0.354    -0.354    -0.354    -0.354  
## genN72-3058:locPlymouth -0.354     -0.354    -0.354    -0.354    -0.354  
## genN72-3148:locPlymouth -0.354     -0.354    -0.354    -0.354    -0.354  
## genN73-1102:locPlymouth -0.707     -0.354    -0.354    -0.354    -0.354  
## genN73-693:locPlymouth  -0.354     -0.707    -0.354    -0.354    -0.354  
## genN73-877:locPlymouth  -0.354     -0.354    -0.707    -0.354    -0.354  
## genN73-882:locPlymouth  -0.354     -0.354    -0.354    -0.707    -0.354  
## genR73-81:locPlymouth   -0.354     -0.354    -0.354    -0.354    -0.707  
## genR75-12:locPlymouth   -0.354     -0.354    -0.354    -0.354    -0.354  
## genTracy:locPlymouth    -0.354     -0.354    -0.354    -0.354    -0.354  
##                         gnR75-12 gnTrcy lcClnt lcPlym gD74-7741:C gN72-137:C
## genD74-7741                                                                 
## genN72-137                                                                  
## genN72-3058                                                                 
## genN72-3148                                                                 
## genN73-1102                                                                 
## genN73-693                                                                  
## genN73-877                                                                  
## genN73-882                                                                  
## genR73-81                                                                   
## genR75-12                                                                   
## genTracy                 0.500                                              
## locClinton               0.500    0.500                                     
## locPlymouth              0.500    0.500  0.500                              
## genD74-7741:locClinton  -0.354   -0.354 -0.707 -0.354                       
## genN72-137:locClinton   -0.354   -0.354 -0.707 -0.354  0.500                
## genN72-3058:locClinton  -0.354   -0.354 -0.707 -0.354  0.500       0.500    
## genN72-3148:locClinton  -0.354   -0.354 -0.707 -0.354  0.500       0.500    
## genN73-1102:locClinton  -0.354   -0.354 -0.707 -0.354  0.500       0.500    
## genN73-693:locClinton   -0.354   -0.354 -0.707 -0.354  0.500       0.500    
## genN73-877:locClinton   -0.354   -0.354 -0.707 -0.354  0.500       0.500    
## genN73-882:locClinton   -0.354   -0.354 -0.707 -0.354  0.500       0.500    
## genR73-81:locClinton    -0.354   -0.354 -0.707 -0.354  0.500       0.500    
## genR75-12:locClinton    -0.707   -0.354 -0.707 -0.354  0.500       0.500    
## genTracy:locClinton     -0.354   -0.707 -0.707 -0.354  0.500       0.500    
## genD74-7741:locPlymouth -0.354   -0.354 -0.354 -0.707  0.500       0.250    
## genN72-137:locPlymouth  -0.354   -0.354 -0.354 -0.707  0.250       0.500    
## genN72-3058:locPlymouth -0.354   -0.354 -0.354 -0.707  0.250       0.250    
## genN72-3148:locPlymouth -0.354   -0.354 -0.354 -0.707  0.250       0.250    
## genN73-1102:locPlymouth -0.354   -0.354 -0.354 -0.707  0.250       0.250    
## genN73-693:locPlymouth  -0.354   -0.354 -0.354 -0.707  0.250       0.250    
## genN73-877:locPlymouth  -0.354   -0.354 -0.354 -0.707  0.250       0.250    
## genN73-882:locPlymouth  -0.354   -0.354 -0.354 -0.707  0.250       0.250    
## genR73-81:locPlymouth   -0.354   -0.354 -0.354 -0.707  0.250       0.250    
## genR75-12:locPlymouth   -0.707   -0.354 -0.354 -0.707  0.250       0.250    
## genTracy:locPlymouth    -0.354   -0.707 -0.354 -0.707  0.250       0.250    
##                         gN72-3058:C gN72-3148:C gN73-1102:C gN73-693:C
## genD74-7741                                                           
## genN72-137                                                            
## genN72-3058                                                           
## genN72-3148                                                           
## genN73-1102                                                           
## genN73-693                                                            
## genN73-877                                                            
## genN73-882                                                            
## genR73-81                                                             
## genR75-12                                                             
## genTracy                                                              
## locClinton                                                            
## locPlymouth                                                           
## genD74-7741:locClinton                                                
## genN72-137:locClinton                                                 
## genN72-3058:locClinton                                                
## genN72-3148:locClinton   0.500                                        
## genN73-1102:locClinton   0.500       0.500                            
## genN73-693:locClinton    0.500       0.500       0.500                
## genN73-877:locClinton    0.500       0.500       0.500       0.500    
## genN73-882:locClinton    0.500       0.500       0.500       0.500    
## genR73-81:locClinton     0.500       0.500       0.500       0.500    
## genR75-12:locClinton     0.500       0.500       0.500       0.500    
## genTracy:locClinton      0.500       0.500       0.500       0.500    
## genD74-7741:locPlymouth  0.250       0.250       0.250       0.250    
## genN72-137:locPlymouth   0.250       0.250       0.250       0.250    
## genN72-3058:locPlymouth  0.500       0.250       0.250       0.250    
## genN72-3148:locPlymouth  0.250       0.500       0.250       0.250    
## genN73-1102:locPlymouth  0.250       0.250       0.500       0.250    
## genN73-693:locPlymouth   0.250       0.250       0.250       0.500    
## genN73-877:locPlymouth   0.250       0.250       0.250       0.250    
## genN73-882:locPlymouth   0.250       0.250       0.250       0.250    
## genR73-81:locPlymouth    0.250       0.250       0.250       0.250    
## genR75-12:locPlymouth    0.250       0.250       0.250       0.250    
## genTracy:locPlymouth     0.250       0.250       0.250       0.250    
##                         gN73-877:C gN73-882:C gR73-81:C gR75-12:C gnTr:C
## genD74-7741                                                             
## genN72-137                                                              
## genN72-3058                                                             
## genN72-3148                                                             
## genN73-1102                                                             
## genN73-693                                                              
## genN73-877                                                              
## genN73-882                                                              
## genR73-81                                                               
## genR75-12                                                               
## genTracy                                                                
## locClinton                                                              
## locPlymouth                                                             
## genD74-7741:locClinton                                                  
## genN72-137:locClinton                                                   
## genN72-3058:locClinton                                                  
## genN72-3148:locClinton                                                  
## genN73-1102:locClinton                                                  
## genN73-693:locClinton                                                   
## genN73-877:locClinton                                                   
## genN73-882:locClinton    0.500                                          
## genR73-81:locClinton     0.500      0.500                               
## genR75-12:locClinton     0.500      0.500      0.500                    
## genTracy:locClinton      0.500      0.500      0.500     0.500          
## genD74-7741:locPlymouth  0.250      0.250      0.250     0.250     0.250
## genN72-137:locPlymouth   0.250      0.250      0.250     0.250     0.250
## genN72-3058:locPlymouth  0.250      0.250      0.250     0.250     0.250
## genN72-3148:locPlymouth  0.250      0.250      0.250     0.250     0.250
## genN73-1102:locPlymouth  0.250      0.250      0.250     0.250     0.250
## genN73-693:locPlymouth   0.250      0.250      0.250     0.250     0.250
## genN73-877:locPlymouth   0.500      0.250      0.250     0.250     0.250
## genN73-882:locPlymouth   0.250      0.500      0.250     0.250     0.250
## genR73-81:locPlymouth    0.250      0.250      0.500     0.250     0.250
## genR75-12:locPlymouth    0.250      0.250      0.250     0.500     0.250
## genTracy:locPlymouth     0.250      0.250      0.250     0.250     0.500
##                         gD74-7741:P gN72-137:P gN72-3058:P gN72-3148:P
## genD74-7741                                                           
## genN72-137                                                            
## genN72-3058                                                           
## genN72-3148                                                           
## genN73-1102                                                           
## genN73-693                                                            
## genN73-877                                                            
## genN73-882                                                            
## genR73-81                                                             
## genR75-12                                                             
## genTracy                                                              
## locClinton                                                            
## locPlymouth                                                           
## genD74-7741:locClinton                                                
## genN72-137:locClinton                                                 
## genN72-3058:locClinton                                                
## genN72-3148:locClinton                                                
## genN73-1102:locClinton                                                
## genN73-693:locClinton                                                 
## genN73-877:locClinton                                                 
## genN73-882:locClinton                                                 
## genR73-81:locClinton                                                  
## genR75-12:locClinton                                                  
## genTracy:locClinton                                                   
## genD74-7741:locPlymouth                                               
## genN72-137:locPlymouth   0.500                                        
## genN72-3058:locPlymouth  0.500       0.500                            
## genN72-3148:locPlymouth  0.500       0.500      0.500                 
## genN73-1102:locPlymouth  0.500       0.500      0.500       0.500     
## genN73-693:locPlymouth   0.500       0.500      0.500       0.500     
## genN73-877:locPlymouth   0.500       0.500      0.500       0.500     
## genN73-882:locPlymouth   0.500       0.500      0.500       0.500     
## genR73-81:locPlymouth    0.500       0.500      0.500       0.500     
## genR75-12:locPlymouth    0.500       0.500      0.500       0.500     
## genTracy:locPlymouth     0.500       0.500      0.500       0.500     
##                         gN73-1102:P gN73-693:P gN73-877:P gN73-882:P gR73-81:P
## genD74-7741                                                                   
## genN72-137                                                                    
## genN72-3058                                                                   
## genN72-3148                                                                   
## genN73-1102                                                                   
## genN73-693                                                                    
## genN73-877                                                                    
## genN73-882                                                                    
## genR73-81                                                                     
## genR75-12                                                                     
## genTracy                                                                      
## locClinton                                                                    
## locPlymouth                                                                   
## genD74-7741:locClinton                                                        
## genN72-137:locClinton                                                         
## genN72-3058:locClinton                                                        
## genN72-3148:locClinton                                                        
## genN73-1102:locClinton                                                        
## genN73-693:locClinton                                                         
## genN73-877:locClinton                                                         
## genN73-882:locClinton                                                         
## genR73-81:locClinton                                                          
## genR75-12:locClinton                                                          
## genTracy:locClinton                                                           
## genD74-7741:locPlymouth                                                       
## genN72-137:locPlymouth                                                        
## genN72-3058:locPlymouth                                                       
## genN72-3148:locPlymouth                                                       
## genN73-1102:locPlymouth                                                       
## genN73-693:locPlymouth   0.500                                                
## genN73-877:locPlymouth   0.500       0.500                                    
## genN73-882:locPlymouth   0.500       0.500      0.500                         
## genR73-81:locPlymouth    0.500       0.500      0.500      0.500              
## genR75-12:locPlymouth    0.500       0.500      0.500      0.500      0.500   
## genTracy:locPlymouth     0.500       0.500      0.500      0.500      0.500   
##                         gR75-12:P
## genD74-7741                      
## genN72-137                       
## genN72-3058                      
## genN72-3148                      
## genN73-1102                      
## genN73-693                       
## genN73-877                       
## genN73-882                       
## genR73-81                        
## genR75-12                        
## genTracy                         
## locClinton                       
## locPlymouth                      
## genD74-7741:locClinton           
## genN72-137:locClinton            
## genN72-3058:locClinton           
## genN72-3148:locClinton           
## genN73-1102:locClinton           
## genN73-693:locClinton            
## genN73-877:locClinton            
## genN73-882:locClinton            
## genR73-81:locClinton             
## genR75-12:locClinton             
## genTracy:locClinton              
## genD74-7741:locPlymouth          
## genN72-137:locPlymouth           
## genN72-3058:locPlymouth          
## genN72-3148:locPlymouth          
## genN73-1102:locPlymouth          
## genN73-693:locPlymouth           
## genN73-877:locPlymouth           
## genN73-882:locPlymouth           
## genR73-81:locPlymouth            
## genR75-12:locPlymouth            
## genTracy:locPlymouth     0.500   
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -1.98153540 -0.61983393  0.03020635  0.55700474  2.01536648 
## 
## Number of Observations: 108
## Number of Groups: 3

No significant interaction effect (p-value = 0.2211), we do find a significant genotype (p-value = 0.0462) and location (p-value = 0.0038) effect.

e.) Regardless of your result in c.), fit a model without interaction effect and then pairwisely compare all the genotypes based on this model. Give your R code to do all of this and report all genotypes which are significantly different from genotype N72-3058.

soy.lme.add = lme(yield~gen+loc, random=~1|block, data=soy)

library(emmeans)
soy.add.emm = emmeans(soy.lme.add, pairwise~gen)
summary(soy.add.emm)
## $emmeans
##  gen        emmean   SE df lower.CL upper.CL
##  Centennial   1395 47.4  2     1191     1599
##  D74-7741     1406 47.4  2     1202     1611
##  N72-137      1484 47.4  2     1280     1688
##  N72-3058     1331 47.4  2     1127     1535
##  N72-3148     1566 47.4  2     1362     1770
##  N73-1102     1501 47.4  2     1297     1706
##  N73-693      1436 47.4  2     1232     1640
##  N73-877      1403 47.4  2     1199     1607
##  N73-882      1397 47.4  2     1193     1601
##  R73-81       1374 47.4  2     1170     1578
##  R75-12       1176 47.4  2      972     1380
##  Tracy        1370 47.4  2     1166     1574
## 
## Results are averaged over the levels of: loc 
## Degrees-of-freedom method: containment 
## Confidence level used: 0.95 
## 
## $contrasts
##  contrast                estimate   SE df t.ratio p.value
##  Centennial - (D74-7741)   -11.67 67.1 92 -0.174  1.0000 
##  Centennial - (N72-137)    -89.11 67.1 92 -1.329  0.9731 
##  Centennial - (N72-3058)    64.11 67.1 92  0.956  0.9982 
##  Centennial - (N72-3148)  -171.22 67.1 92 -2.553  0.3219 
##  Centennial - (N73-1102)  -106.67 67.1 92 -1.590  0.9082 
##  Centennial - (N73-693)    -41.44 67.1 92 -0.618  1.0000 
##  Centennial - (N73-877)     -8.56 67.1 92 -0.128  1.0000 
##  Centennial - (N73-882)     -1.89 67.1 92 -0.028  1.0000 
##  Centennial - (R73-81)      21.00 67.1 92  0.313  1.0000 
##  Centennial - (R75-12)     218.89 67.1 92  3.264  0.0640 
##  Centennial - Tracy         24.89 67.1 92  0.371  1.0000 
##  (D74-7741) - (N72-137)    -77.44 67.1 92 -1.155  0.9910 
##  (D74-7741) - (N72-3058)    75.78 67.1 92  1.130  0.9925 
##  (D74-7741) - (N72-3148)  -159.56 67.1 92 -2.379  0.4306 
##  (D74-7741) - (N73-1102)   -95.00 67.1 92 -1.416  0.9573 
##  (D74-7741) - (N73-693)    -29.78 67.1 92 -0.444  1.0000 
##  (D74-7741) - (N73-877)      3.11 67.1 92  0.046  1.0000 
##  (D74-7741) - (N73-882)      9.78 67.1 92  0.146  1.0000 
##  (D74-7741) - (R73-81)      32.67 67.1 92  0.487  1.0000 
##  (D74-7741) - (R75-12)     230.56 67.1 92  3.437  0.0394 
##  (D74-7741) - Tracy         36.56 67.1 92  0.545  1.0000 
##  (N72-137) - (N72-3058)    153.22 67.1 92  2.284  0.4949 
##  (N72-137) - (N72-3148)    -82.11 67.1 92 -1.224  0.9856 
##  (N72-137) - (N73-1102)    -17.56 67.1 92 -0.262  1.0000 
##  (N72-137) - (N73-693)      47.67 67.1 92  0.711  0.9999 
##  (N72-137) - (N73-877)      80.56 67.1 92  1.201  0.9876 
##  (N72-137) - (N73-882)      87.22 67.1 92  1.300  0.9771 
##  (N72-137) - (R73-81)      110.11 67.1 92  1.642  0.8887 
##  (N72-137) - (R75-12)      308.00 67.1 92  4.592  0.0008 
##  (N72-137) - Tracy         114.00 67.1 92  1.700  0.8638 
##  (N72-3058) - (N72-3148)  -235.33 67.1 92 -3.509  0.0320 
##  (N72-3058) - (N73-1102)  -170.78 67.1 92 -2.546  0.3257 
##  (N72-3058) - (N73-693)   -105.56 67.1 92 -1.574  0.9140 
##  (N72-3058) - (N73-877)    -72.67 67.1 92 -1.083  0.9947 
##  (N72-3058) - (N73-882)    -66.00 67.1 92 -0.984  0.9977 
##  (N72-3058) - (R73-81)     -43.11 67.1 92 -0.643  1.0000 
##  (N72-3058) - (R75-12)     154.78 67.1 92  2.308  0.4789 
##  (N72-3058) - Tracy        -39.22 67.1 92 -0.585  1.0000 
##  (N72-3148) - (N73-1102)    64.56 67.1 92  0.962  0.9981 
##  (N72-3148) - (N73-693)    129.78 67.1 92  1.935  0.7344 
##  (N72-3148) - (N73-877)    162.67 67.1 92  2.425  0.4002 
##  (N72-3148) - (N73-882)    169.33 67.1 92  2.525  0.3384 
##  (N72-3148) - (R73-81)     192.22 67.1 92  2.866  0.1710 
##  (N72-3148) - (R75-12)     390.11 67.1 92  5.816  <.0001 
##  (N72-3148) - Tracy        196.11 67.1 92  2.924  0.1500 
##  (N73-1102) - (N73-693)     65.22 67.1 92  0.972  0.9979 
##  (N73-1102) - (N73-877)     98.11 67.1 92  1.463  0.9467 
##  (N73-1102) - (N73-882)    104.78 67.1 92  1.562  0.9180 
##  (N73-1102) - (R73-81)     127.67 67.1 92  1.903  0.7540 
##  (N73-1102) - (R75-12)     325.56 67.1 92  4.854  0.0003 
##  (N73-1102) - Tracy        131.56 67.1 92  1.961  0.7175 
##  (N73-693) - (N73-877)      32.89 67.1 92  0.490  1.0000 
##  (N73-693) - (N73-882)      39.56 67.1 92  0.590  1.0000 
##  (N73-693) - (R73-81)       62.44 67.1 92  0.931  0.9986 
##  (N73-693) - (R75-12)      260.33 67.1 92  3.881  0.0100 
##  (N73-693) - Tracy          66.33 67.1 92  0.989  0.9976 
##  (N73-877) - (N73-882)       6.67 67.1 92  0.099  1.0000 
##  (N73-877) - (R73-81)       29.56 67.1 92  0.441  1.0000 
##  (N73-877) - (R75-12)      227.44 67.1 92  3.391  0.0450 
##  (N73-877) - Tracy          33.44 67.1 92  0.499  1.0000 
##  (N73-882) - (R73-81)       22.89 67.1 92  0.341  1.0000 
##  (N73-882) - (R75-12)      220.78 67.1 92  3.292  0.0593 
##  (N73-882) - Tracy          26.78 67.1 92  0.399  1.0000 
##  (R73-81) - (R75-12)       197.89 67.1 92  2.950  0.1410 
##  (R73-81) - Tracy            3.89 67.1 92  0.058  1.0000 
##  (R75-12) - Tracy         -194.00 67.1 92 -2.892  0.1611 
## 
## Results are averaged over the levels of: loc 
## Degrees-of-freedom method: containment 
## P value adjustment: tukey method for comparing a family of 12 estimates

The only significant difference is between genotype N72-3058 and N72-3148 (p-value = 0.0320). #### f.) According to the model from e.), how many kg of yield do we obtain for the Centennial genotype, grown in Plymouth, on average? Justify your answer.

fixef(summary(soy.lme.add))
## (Intercept) genD74-7741  genN72-137 genN72-3058 genN72-3148 genN73-1102 
## 1193.583333   11.666667   89.111111  -64.111111  171.222222  106.666667 
##  genN73-693  genN73-877  genN73-882   genR73-81   genR75-12    genTracy 
##   41.444444    8.555556    1.888889  -21.000000 -218.888889  -24.888889 
##  locClinton locPlymouth 
##  415.305556  188.277778
fixef(summary(soy.lme.add))['(Intercept)'] + fixef(summary(soy.lme.add))['locPlymouth']
## (Intercept) 
##    1381.861

The average yield of the Centennial genotype grown in Plymouth is 1381.861 kg.

g.) How high is the variance of the yield values, and what proportion of it is explained with the model from e.)? Explain your approach.

yield.var = var(soy$yield)
library(MuMIn)
r.squaredGLMM(soy.lme.add)
##            R2m       R2c
## [1,] 0.6509061 0.6509061

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Exam 2019

Problem 1 - the federer tobacco data (one and two factorial experiment with block structure)

library(agridat)
data('federer.tobacco')
str(federer.tobacco)
## 'data.frame':    56 obs. of  4 variables:
##  $ row   : int  1 1 1 1 1 1 1 1 2 2 ...
##  $ block : int  1 2 3 4 5 6 7 8 1 2 ...
##  $ dose  : int  2500 250 0 2500 2500 5000 250 1000 5000 1500 ...
##  $ height: num  1299 1369 1170 1219 1120 ...
federer.tobacco$dose = factor(federer.tobacco$dose)
federer.tobacco$row = factor(federer.tobacco$row)
federer.tobacco$block = factor(federer.tobacco$block)

a.) Give the R code to fit a suitable parametric model to this data set to answer the research question.

library(nlme)
tob.lme = lme(height~dose, random=~1|block, data=federer.tobacco)

b.) Is the dose effect significant according to the model from a.)? Name the tools used, give the R code to perform the tests, report the p value and use it to answer the question.(2)

anova(tob.lme, type='marginal')

I performed an overall F test and we get that the dose effect is not significant (p-value = 0.1985)

c.) The layout of the experiment was in eight rows and seven columns. Shortly after the plants were transplanted to the field it became apparent that an environmental gradient existed. The following residual plot for the model in a.) was produced: Use this residual plot to explain that there is a gradient and discuss what it looks like.(2)

The residuals are observed values − fitted values. Accordingly, the rows 2, 3, and 4 (with negative residuals) seem to produce systematically lower values than the rows 1,5, and 7.

d.) To take care of the mentioned gradient, one can add the fixed effect of the row to the model. Give the R code to fit this model.

tob.lme.row = lme(height~dose + row, random=~1|block, data=federer.tobacco)

e.) According to the model in d.), is the radiation effect significant? And the row effect? Use the same tools as in b.), give your R code, p values and your decision. (1)

anova(tob.lme.row, type='marginal')

When accounting for the fixed effect of the row we get a significant row effect (p-value=<.0001) and a significant dose effect (p-value=0.0281).

f.) According to the model in d.), by how many cm does the total plant height (of the 20 plants) decrease on average for a “typical” block if the dose was 5’000 roentgen, compared to 0 roentgen? Explain your approach.

fixef(tob.lme.row)
## (Intercept)     dose250     dose500    dose1000    dose1500    dose2500 
## 1185.530716   23.070073   21.832451    2.666409  -32.124437   -6.580593 
##    dose5000        row2        row3        row4        row5        row6 
## -128.810507 -243.365185 -400.603993 -251.300986  -47.647904 -129.270664 
##        row7 
##  -19.254680
fixef(tob.lme.row)['dose5000']
##  dose5000 
## -128.8105

The dose is fit as a reference level, we see that the average height decreases 128.8105 units for a typical block for a dose of 5000.

g.) Is the answer to f.) different for different rows or not? Why or why not?

No it is not different for different rows since we do not take interaction effects between dose and row in consideration.

h.) Assess the model assumptions: explain what you assess, with which method, give your R code, discuss the results (give p values where you calculate them) and report your conclusions regarding the assumptions.

library(car)
qqPlot(resid(tob.lme.row))

##  1  2 
## 49 10
plot(tob.lme.row)

We assess the normality assumption and the assumption of equal variance for the residuals, we do not check normality of the blocks since we only have 8 different blocks.

The qqPlot of the residuals does not show any problems with normality, the variance of the residuals seems to increase with the fitted values. This could be problematic.

i) You should have found a problem with the assumptions. What could you do about it? Give a precise answer for full points, not only a rough idea. (2)

We found problems with the variance, since we do not have problems with normality we do not want to transform the data but instead model heteroscedastic within group errors.

tob.lme.het = lme(height~dose+row, random=~1|block, data=federer.tobacco, weights=varIdent(form=~1|row))
plot(tob.lme.het)

j.) We want to compare each of the dose levels only to its adjacent levels (0 to 250, 250 to 500, . . . ). Show how to test which of these differences are significant, and give the significant results only, with p values.

library(emmeans)

tob.emm = emmeans(tob.lme.row, pairwise~dose)
tob.emm
## $emmeans
##  dose emmean   SE df lower.CL upper.CL
##  0      1030 43.8  7      926     1133
##  250    1053 43.2  7      951     1155
##  500    1051 44.6  7      946     1157
##  1000   1032 44.6  7      927     1138
##  1500    997 44.1  7      893     1102
##  2500   1023 43.6  7      920     1126
##  5000    901 42.7  7      800     1002
## 
## Results are averaged over the levels of: row 
## Degrees-of-freedom method: containment 
## Confidence level used: 0.95 
## 
## $contrasts
##  contrast    estimate   SE df t.ratio p.value
##  0 - 250       -23.07 44.1 36 -0.523  0.9983 
##  0 - 500       -21.83 45.4 36 -0.481  0.9990 
##  0 - 1000       -2.67 49.9 36 -0.053  1.0000 
##  0 - 1500       32.12 46.9 36  0.685  0.9927 
##  0 - 2500        6.58 47.2 36  0.140  1.0000 
##  0 - 5000      128.81 45.5 36  2.831  0.0966 
##  250 - 500       1.24 47.0 36  0.026  1.0000 
##  250 - 1000     20.40 48.1 36  0.424  0.9995 
##  250 - 1500     55.19 46.2 36  1.195  0.8913 
##  250 - 2500     29.65 44.6 36  0.666  0.9937 
##  250 - 5000    151.88 45.2 36  3.359  0.0280 
##  500 - 1000     19.17 47.8 36  0.401  0.9996 
##  500 - 1500     53.96 49.7 36  1.087  0.9280 
##  500 - 2500     28.41 48.4 36  0.588  0.9968 
##  500 - 5000    150.64 46.3 36  3.251  0.0366 
##  1000 - 1500    34.79 47.0 36  0.740  0.9890 
##  1000 - 2500     9.25 46.1 36  0.200  1.0000 
##  1000 - 5000   131.48 45.7 36  2.874  0.0878 
##  1500 - 2500   -25.54 46.8 36 -0.546  0.9979 
##  1500 - 5000    96.69 44.5 36  2.175  0.3336 
##  2500 - 5000   122.23 44.8 36  2.726  0.1207 
## 
## Results are averaged over the levels of: row 
## Degrees-of-freedom method: containment 
## P value adjustment: tukey method for comparing a family of 7 estimates
library(multcomp)
summary(glht(tob.lme.row, mcp(dose='Sequen')))
## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Multiple Comparisons of Means: Sequen Contrasts
## 
## 
## Fit: lme.formula(fixed = height ~ dose + row, data = federer.tobacco, 
##     random = ~1 | block)
## 
## Linear Hypotheses:
##                  Estimate Std. Error z value Pr(>|z|)  
## 250 - 0 == 0       23.070     44.140   0.523   0.9913  
## 500 - 250 == 0     -1.238     47.047  -0.026   1.0000  
## 1000 - 500 == 0   -19.166     47.811  -0.401   0.9978  
## 1500 - 1000 == 0  -34.791     47.044  -0.740   0.9560  
## 2500 - 1500 == 0   25.544     46.761   0.546   0.9892  
## 5000 - 2500 == 0 -122.230     44.833  -2.726   0.0356 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)

0 - 250 -23.07 44.1 36 -0.523 0.9983 250 - 500 1.24 47.0 36 0.026 1.0000 250 - 500 1.24 47.0 36 0.026 1.0000 1000 - 1500 34.79 47.0 36 0.740 0.9890 1500 - 2500 -25.54 46.8 36 -0.546 0.9979 2500 - 5000 122.23 44.8 36 2.726 0.1207

None of the comparisons are significant according to the emmeans method since it makes more comparisons and is thus stricter controling the familywise error rate at 5%. When comparing with the ‘Sequen’ method of the multcomp package we do get a significant difference for the difference of level 2500 and 5000 with a p-value of 0.0354. This method has a less strict p-value correction.

Problem 2 - the cotton data (multifactorial experiment, no blocks)

library(agridat)
data('gregory.cotton')
str(gregory.cotton)
## 'data.frame':    144 obs. of  6 variables:
##  $ yield   : num  0.99 1.34 1.26 1.44 1.4 1.36 1.23 1.28 1.56 1.64 ...
##  $ year    : Factor w/ 2 levels "Y1","Y2": 1 1 1 1 1 1 1 1 1 1 ...
##  $ nitrogen: Factor w/ 2 levels "N0","N1": 1 1 1 1 1 1 1 1 1 1 ...
##  $ date    : Factor w/ 4 levels "D1","D2","D3",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ water   : Factor w/ 3 levels "I1","I2","I3": 1 1 1 2 2 2 3 3 3 1 ...
##  $ spacing : Factor w/ 3 levels "S1","S2","S3": 1 2 3 1 2 3 1 2 3 1 ...

a.) According to the sample data from year Y1: which of the four dates should we choose to get the highest average yield if we use the nitrogen fertilizer, and how high is the average yield for this sowing date in year Y1 for the fertilized plots? (1)

cotton.Y1 = gregory.cotton[gregory.cotton$year == 'Y1' & gregory.cotton$nitrogen == 'N1', ]
with(cotton.Y1, tapply(yield, date, mean))
##       D1       D2       D3       D4 
## 2.773333 3.074444 2.664444 1.735556

The highest average yield is reported for date D2 with 3.074units.

b.) We begin with a model that has no year effect. Give the R code to model the effect of the sowing date, the nitrogen fertilizer and their interaction on the yield.

cotton.lm = lm(yield~nitrogen*date, data=gregory.cotton)

c.) Does the effect of the nitrogen fertilizer on the yield depend on the date according to the model from b.)? Give your code, and answer the question, argue with p values. (2)

library(car)
Anova(cotton.lm, type=2)

We perform an overall F test and get a significant interaction effect of nitrogen and date (p-value = 0.001748). We conclude that the effect of the nitrogen fertilizer does depend on the date.

d.) How many interaction terms are involved in the interaction effect of this model? (1)

unique(gregory.cotton$nitrogen)
## [1] N0 N1
## Levels: N0 N1
unique(gregory.cotton$date)
## [1] D1 D2 D3 D4
## Levels: D1 D2 D3 D4

We have two levels of nitrogen and four levels for dates, one of the levels each is the reference level, hence we get three interaction terms. (4-1)\(\cdot\)(2-1)=3

e.) According to the interaction plot on the following page, it seems that only a part of the interaction terms may be needed from a statistical point of view. Which one looks the most needed, and why?

The interaction plot is more or less parallel for the interactions of D1, D2 and D3, for D4 the lines are very close together, hence we assume that the interaction between D4 and N1 is the most important term.

f.) Use significance tests for the interaction terms to check your answer to problem e.). Give your code and argue with p values to show that your answer was correct. (2)

summary(cotton.lm)
## 
## Call:
## lm(formula = yield ~ nitrogen * date, data = gregory.cotton)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.9900 -0.3644 -0.1153  0.3664  1.2700 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        1.15389    0.11511  10.024  < 2e-16 ***
## nitrogenN1         1.14167    0.16279   7.013 9.93e-11 ***
## dateD2             0.33278    0.16279   2.044 0.042857 *  
## dateD3             0.25722    0.16279   1.580 0.116401    
## dateD4            -0.01944    0.16279  -0.119 0.905097    
## nitrogenN1:dateD2 -0.04667    0.23021  -0.203 0.839665    
## nitrogenN1:dateD3 -0.33278    0.23021  -1.446 0.150613    
## nitrogenN1:dateD4 -0.81556    0.23021  -3.543 0.000543 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4884 on 136 degrees of freedom
## Multiple R-squared:  0.5502, Adjusted R-squared:  0.5271 
## F-statistic: 23.77 on 7 and 136 DF,  p-value: < 2.2e-16

The interaction terms for nitrogen level N1 and date level D2 as well as between nitrogen level N1 and date level D3 are not significant, with p-values 0.839665 and 0.150613 respectively. We do have a significant interaction term for nitrogen level N1 and date level D4 with p-value = 0.000543.

g.) We now start to investigate the effect of the year. Add the missing yield values for date D4 in year Y2 with nitrogen level N1 to the plot below.

library(ggplot2)
ggplot(gregory.cotton, aes(date, yield)) + geom_point() + facet_grid(year~nitrogen)

h.) Add the year to the model. Keep in mind that the year effect could also depend on the levels of the nitrogen and the date. Give your R code to fit a suitable model. (1)

cotton.lm.year = lm(yield~nitrogen*date*year, gregory.cotton)

i.) Refer to your model from h.). Which (if any) of the interaction effects with the year is significant? Give your R code and argue with p values. (1)

library(car)
Anova(cotton.lm.year, type=2)

The nitrogen and year interaction effect is significant with a p-value of 0.02743. We conclude that the effect of the nitrogen on the yield depends on the year.

j.) According to your model in h.) (without removing any nonsignificant effects), how high is the average yield for a plot with N1 nitrogen level, at date D4 in year Y2? Give your code and your answer.

coef(summary(cotton.lm.year))
##                            Estimate Std. Error   t value     Pr(>|t|)
## (Intercept)               1.3177778 0.09949021 13.245301 9.397591e-26
## nitrogenN1                1.4555556 0.14070041 10.345070 1.333739e-18
## dateD2                    0.5711111 0.14070041  4.059058 8.522107e-05
## dateD3                    0.4800000 0.14070041  3.411504 8.650602e-04
## dateD4                    0.1100000 0.14070041  0.781803 4.357741e-01
## yearY2                   -0.3277778 0.14070041 -2.329615 2.139219e-02
## nitrogenN1:dateD2        -0.2700000 0.19898043 -1.356917 1.771956e-01
## nitrogenN1:dateD3        -0.5888889 0.19898043 -2.959532 3.671954e-03
## nitrogenN1:dateD4        -1.1477778 0.19898043 -5.768295 5.696978e-08
## nitrogenN1:yearY2        -0.6277778 0.19898043 -3.154973 2.000966e-03
## dateD2:yearY2            -0.4766667 0.19898043 -2.395546 1.804359e-02
## dateD3:yearY2            -0.4455556 0.19898043 -2.239193 2.686976e-02
## dateD4:yearY2            -0.2588889 0.19898043 -1.301077 1.955692e-01
## nitrogenN1:dateD2:yearY2  0.4466667 0.28140082  1.587297 1.149127e-01
## nitrogenN1:dateD3:yearY2  0.5122222 0.28140082  1.820258 7.105594e-02
## nitrogenN1:dateD4:yearY2  0.6644444 0.28140082  2.361203 1.972497e-02
coef(summary(cotton.lm.year))['(Intercept)', 'Estimate'] + coef(summary(cotton.lm.year))['nitrogenN1', 'Estimate'] + coef(summary(cotton.lm.year))['dateD4', 'Estimate'] + coef(summary(cotton.lm.year))['yearY2', 'Estimate'] + coef(summary(cotton.lm.year))['nitrogenN1:dateD4', 'Estimate'] + coef(summary(cotton.lm.year))['dateD4:yearY2', 'Estimate'] + coef(summary(cotton.lm.year))['nitrogenN1:dateD4:yearY2', 'Estimate']
## [1] 1.813333
#or 

df.pred = data.frame(date = "D4", nitrogen = "N1", year = "Y2")
predict(cotton.lm.year, df.pred)
##        1 
## 1.185556

The average yield is 1.813333.

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